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Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras
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Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras$ Jianlian Cuia,1 and Jinchuan Houb,,2 b
a School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China Department of Mathematics, Shanxi Teachers University, Linfen, 041004, People’s Republic of China
Received 30 September 2002; revised 22 December 2002; accepted 15 January 2003 Communicated by G. Pisier
Abstract A nonlinear map f between operator algebras is said to be a numerical radius isometry if wðfðT SÞÞ ¼ wðT SÞ for all T; S in its domain algebra, where wðTÞ stands for the numerical radius of T: Let N and M be two atomic nests on complex Hilbert spaces H and K; respectively. Denote Alg N the nest algebra associated with N and DN ¼ Alg N-ðAlg NÞ the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from Alg N onto Alg M and a thorough classification of numerical radius isometries from DN onto DM : r 2003 Elsevier Inc. All rights reserved. MSC: Primary 47H20; 47L35; 47A12; Secondary 47B48 Keywords: Numerical radius; Non-linear isometries; Nest algebras
1. Introduction Let H and K be complex Hilbert spaces with inner product / ; S; and BðH; KÞ (BðHÞ if H ¼ K) be Banach space of all bounded linear operators from H into K: Assume that A and B are subalgebras (or subspaces) in BðHÞ and BðKÞ; $
This work was partly supported by NNSFC and PNSFS. Corresponding author. E-mail address: [email protected] (J. Hou). 1 Department of Applied Mathematics, Taiyuan University of Technology, Taiyuan 030024, People’s Republic of China; Department of Mathematics, Shanxi Teachers University, Linfen, 041004, People’s Republic of China. 2 Department of Mathematics, Shanxi University, Taiyuan 030000, People’s Republic of China.
0022-1236/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-1236(03)00076-4
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respectively, and F : A-B is a map (maybe non-linear). A general and interesting problem that should be considered is as follows: Problem 0.1. Find as few as possible properties that may be possessed by A and B or by elements in them and that are enough to determine the structure of the map F if F takes these properties as invariants, i.e., if F preserves these properties. When F is linear, the above problem is so-called the linear preserver problem, of which the study can be traced back to the work of Frobenius [4] and has become one of the most active and fertile research fields in the matrix theory and operator theory during the past decades (see the survey paper [13]). Many results concerning linear preserver problems reveal the relations between linear structure and algebraic structure of operator algebras. Problem 0.1 is also associated with the geometry of matrices whose study was pioneered by Hua in the 1940s (see [7–10]). Let F be a field and Mnn ðFÞ be the space of all n n matrices over F: Consider the group of motions on Mnn ðFÞ consisting of the following maps: T/PTQ þ R;
8TAMnn ðFÞ
or
T/PT tr Q þ R;
8TAMnn ðFÞ;
where P; Q and R are n n matrices with P and Q being nonsingular, T tr denotes the transpose matrix of T: The fundamental problem of geometry of matrices is to characterize the group of motions (to within automorphisms of the underlying field) by as few geometry invariants as possible [15]. Hua proved that ‘‘adjacency’’ (T and S are adjacent if rankðT SÞ ¼ 1) is such an invariant for Mnn ðFÞ over some field F; especially the real field R and the complex field C: Thus our Problem 0.1 asks indeed the possibility of developing an analog of ‘‘geometry of matrices’’ for operators. The numerical range and numerical radius of TABðHÞ are respectively defined as W ðTÞ ¼ f/Tx; xS j xAH and jjxjj ¼ 1g; wðTÞ ¼ supfjlj j lAW ðTÞg: These concepts and their generalizations have been studied extensively because of their connections and applications to many different areas. Bai and Hou [1] found that the geometry invariant ‘‘numerical radius distance’’ alone is sufficient to characterize the nonlinear maps from BðHÞ onto BðKÞ: Precisely, it was shown that F : BðHÞ-BðKÞ is a surjective numerical radius isometry, i.e., wðFðTÞ FðSÞÞ ¼ wðT SÞ holds for all T; SABðHÞ if and only if there exist complex unit m (i.e., mAC and jmj ¼ 1) and operator RABðKÞ such that F takes one of the following (four) forms: (1) There exists a unitary or conjugate unitary operator U : H-K such that FðAÞ ¼ mUAU þ R for every AABðHÞ:
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(2) There exists a unitary or conjugate unitary operator U : H-K such that FðAÞ ¼ mUA U þ R for every AABðHÞ: Therefore the group of all nonlinear surjective numerical radius isometries on BðHÞ is generated by the following five kinds of very simple maps (motions): (i) (ii) (iii) (iv) (v)
T/UTU ; where U : H-H is a unitary operator; T/UTU ; where U : H-H is a conjugate unitary operator; T/T ; T/cT; where cAC and jcj ¼ 1; T/T þ R; where RABðHÞ:
Note that, it follows from the above result that every (nonlinear) numerical radius isometry from BðHÞ onto BðKÞ is also an operator norm isometry, but the converse proposition fails. In this paper, we consider the same question of characterizing the nonlinear surjective numerical radius isometries for more general cases, i.e., for the nest algebra case and the diagonal algebra case. Let N and M be atomic nests on H and K; respectively. Denote the associated nest algebras by Alg N and Alg M; and the associated diagonal algebras by DN and DM ; respectively. In [2], we characterized the linear maps preserving the numerical radius from DN onto DM and, when N and M are maximal atomic, characterized the weakly continuous linear maps preserving the numerical radius from Alg N onto Alg M: In this paper, we find further that the invariant ‘‘numerical radius distance’’ is also enough to determine the structure of nonlinear maps from DN onto DM and the structure of weakly continuous nonlinear maps from Alg N onto Alg M; and then give their complete classifications. Not like the case of BðHÞ; the group of surjective numerical radius isometries on a nest algebra or a diagonal algebra may has other kinds of generators which are no longer so nicely to keep the algebraic and geometric structures. By Mazur–Ulam theorem [14], to achieve our goal, one of the keys is to reduce the question to that of characterizing the additive maps which preserve the closure of numerical range. For finite dimensional cases, that is, for the cases of upper triangular block matrix algebra T and diagonal block matrix algebra D; the additive maps preserving the numerical range were characterized by Lesnjak [11] and, based on Lesnjak’s work, the classifications of numerical radius isometries on T and D were given by Li and Sˇemrl [12]. The results and the basic ideas in [11,12] are enlightening and useful to our study here. The paper is organized as follows. In Section 2, we give some basic properties of the additive maps which preserve the closure of numerical range from an atomic nest algebra onto another one, and get a characterization of such maps when restricted to the diagonal algebras. Section 3 is the crucial parts of this paper, in which we get a complete description of weakly continuous additive maps preserving the closure of numerical range from a maximal atomic nest algebra onto another one. In Section 4, we generalize the results in Section 3 to the atomic nest algebra case. Based on the discussions in previous sections, Section 5 is devoted to characterizing the numerical
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radius isometries. The main results are Theorems 5.1 and 5.2, which give respectively a complete characterization of the weakly continuous numerical radius isometries between atomic nest algebras and the numerical radius isometries between diagonal algebras. Comparing Theorem 5.1 with Theorem 5.2, we need the additional assumption of weak continuity for maps (even for linear maps, see Theorem 4.2 and [2]) on nest algebras. We guess that the assumption of weak continuity in Theorem 5.1 is not necessary, and we answer affirmatively this conjecture in some situations (See Corollary 5.4). But in this paper we are not able to omit this assumption for general cases. Now we fix some notations. Recall that a nest on H is a chain N of closed (under norm topology) subspaces of H containing f0g and H; which W is closed under the formationV of arbitrary closed linear span (denoted by ) and intersection (denoted by ). Alg N denotes the associated nest algebra, which is the set of all operators T in BðHÞ such that TNDN for every element NAN: We denote AlgF N ¼ Alg N-FðHÞ; the set of all finite rank operators in Alg N: Relation ‘‘ACB’’ means that A is a proper subset of B: The self-adjoint operator algebra DN ¼ Alg N-ðAlg WNÞ is called the diagonalValgebra associated with N: For NAN; define N ¼ fMAN j MCNg; Nþ ¼ fMAN j NCMg and define 0 ¼ 0; Hþ ¼ H: As usual, N > is the orthogonal complement of N: If N~N ¼ N-ðN Þ> a0; we say N~N is an atom of N: A nest N on H is said to be atomic if H is spanned by its atoms, and to be maximal atomic if N is atomic and all its atoms are one dimensional (Ref. [3]). Let x; f AH; the rank-1 operator defined by y//y; f Sx will be denoted by x#f : Two nests N and M on H and K; respectively, is unitarily equivalent if there is a unitary operator (or equivalently, a conjugate unitary operator) U : H-K such that UðNÞ ¼ fUðNÞ j NANg ¼ M: Recall that U is a conjugate unitary operator if U is conjugate linear (i.e., U is additive and UðlxÞ ¼ l% Ux for any lAC and xAH) and both UU and U U are the identity operators. Note that, for a conjugate linear operator A on H; one has /Ax; yS ¼ /A y; xS for all x; yAH: A projection PABðHÞ is a self-adjoint and idempotent operator (i.e., P ¼ P ¼ P2 ). We denote PL the projection whose range is the subspace L: As usual, C; R and N stand for the complex number field, real number field and natural number set, respectively. At the end of introduction, we recall some basic results on numerical range and numerical radius used often in our study. One may see [5] and Chapter 1 of [6] for more information. Proposition 1.1 (See Halmos [5]). Let AABðHÞ: (1) W ðAÞ ¼ W ðAtr Þ; where Atr stands for the transpose of A relative to an arbitrary orthonormal base of H: (2) W ðAÞ ¼ W ðUAU Þ for any unitary UABðHÞ: (3) W ðlAÞ ¼ lW ðAÞ for any lAC: (4) W ðlI þ AÞ ¼ l þ W ðAÞ for any lAC:
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Proposition 1.2 (See Halmos [5]). The numericalrange ofAABðHÞ is always convex. l1 b ; then W ðAÞ is an elliptical In particular, if AAM2 ðCÞ is unitarily similar to 0 l2 disk with l1 and l2 as foci, and the length of minor axis equal to jbj; where M2 ðCÞ denotes the 2 2 complex matrix algebra. Proposition 1.3 (See Halmos [5]). Let AABðHÞ: Then W ðAÞ ¼ flg if and only if A ¼ lI: Proposition 1.4 (See Halmos [5], Horn and Johnson [6]). Suppose that NCH is a closed subspace and AABðHÞ: Then W ðPN AjN ÞDW ðAÞ and wðPN AjN ÞpwðAÞ: Proposition 1.5 (See Horn and Johnson [6, Section 1.2.9]). Suppose that AAMn ðCÞ such that A þ A (resp., AA i ) have ln and l1 as the largest and smallest eigenvalues, respectively. Then n z z o % ½l1 ; ln ¼ fz þ z% j zAW ðAÞg resp:; ½l1 ; ln ¼ zAW ðAÞ : i Proposition 1.6 (See Halmos [5]). If AABðHÞ is unitarily similar to A1 "A2 ; then W ðAÞ ¼ convfW ðA1 Þ,W ðA2 Þg; where convðOÞ denotes the convex hull of the set O:
2. Additive maps preserving the closure of numerical range on diagonal algebras Let N be an atomic nest on Hilbert space H: Assume that fHi j iAJg is the set of all atoms of N; then H ¼ "iAJ Hi : There is a natural order ‘‘p’’ so that J becomes a totally ordered set. For any i; jAJ; ipj if and only if xi #yj AAlg N whenever xi AHi and yj AHj : ioj means ipj and iaj: It is clear in this case that DN ¼ "iAJ BðHi Þ: We say that two projections P; QAAlg N are orthogonal if PQ ¼ 0: Lemma 2.1. Let N and M be two atomic nests on Hilbert spaces H and K; respectively. Assume F : Alg N-Alg M (or F : DN -DM ) is an additive surjective map. If F preserves the closure of numerical range, then F preserves the orthogonal projections of rank one in both directions. Furthermore, FðlPÞ ¼ lFðPÞ for every rank-1 projection P and lAC: Proof. Since F preserves the closure of numerical range, it is easy to see that F is injective and preserves the numerical radius. Note that the numerical radius is a
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norm which is equivalent to the usual operator norm, so F is continuous and hence is real linear. For any finite rank operator F ; we have W ðF Þ ¼ W ðF Þ: Let PAAlg N be an arbitrary rank-1 projection. Then ½0; 1 ¼ W ðPÞ ¼ W ðPÞ ¼ W ðFðPÞÞ; and therefore, FðPÞX0: Set FðPÞ ¼ B þ C; where B and C are positive operators in Alg M; then P ¼ F1 ðBÞ þ F1 ðCÞ; here F1 ðBÞ and F1 ðCÞ are positive since F1 also preserves the closure of numerical range. Lemma 2.1 in [2] states that a positive operator A in a nest algebra is a scalar multiple of a projection of rank one if and only if A ¼ B þ C with B and C being positive in the nest algebra implies that both B and C are scalar multiples of A: Thus, there is a real number rA½0; 1 such that F1 ðBÞ ¼ rP and F1 ðCÞ ¼ ð1 rÞP: So B ¼ rFðPÞ and C ¼ ð1 rÞFðPÞ by the real linearity of F: Consequently, FðPÞ is a positive multiple of a projection of rank one. Since W ðFðPÞÞ ¼ W ðFðPÞÞ ¼ ½0; 1; FðPÞ is a rank-1 projection. Now suppose that rank-1 projections P; QAAlg N are orthogonal. Let FðPÞ ¼ x#x and FðQÞ ¼ y#y: If x and y belong to the different atoms of M; then it is clear that FðPÞ and FðQÞ are orthogonal. Otherwise, there exists MAM with dimðM~M ÞX2 such that x; yAM~M : Since fj/z; xSj2 j/z; ySj2 j jjzjj ¼ 1g ¼ W ðFðP QÞÞ ¼ W ðP QÞ ¼ ½1; 1; and since j/z; xSj2 j/z; ySj2 attains the largest value 1 only when z and x are linearly dependent and z>y; we must have /x; yS ¼ 0: So FðPÞ and FðQÞ are orthogonal. Because F1 has the same properties as F has, F preserves the orthogonal projections of rank one in both directions. Let u#uAAlg N be a projection, Fðu#uÞ ¼ x#x and let Fðiu#uÞ ¼ B; where i denotes the imaginary unit. For any unit vector y in some atom of M such that y>x; there is a unit vector v in some atom of N such that Fðv#vÞ ¼ y#y: Note that v>u: It is clear that, for a sufficiently large positive number M; we have wðiu#u þ Mv#vÞ ¼ M: Thus wðB þ My#yÞ ¼ M as F preserves the numerical radius. It follows that j/By; yS þ MjpM; and consequently, /By; yS ¼ 0 by virtue of iBX0: Therefore, B must be a scalar multiple of x#x: Now it is obvious that Fðiu#uÞ ¼ ix#x and F is complex linear when restricted to the one dimensional space spanned by a rank-1 projection. The proof for the case F : DN -DM is just the same. & Lemma 2.2. Let N and M be two atomic nests on Hilbert spaces H and K; respectively, and let fHj j jAJg and fKk j kAKg be the set of all atoms of N and M; respectively. Assume F : Alg N-Alg M (or F : DN -DM ) is an additive surjective map. If F preserves the closure of numerical range, then there exists a 1-1 and onto map y : J-K such that for every jAJ; FðBðHj ÞÞ ¼ BðKyð jÞ Þ; here we identify BðHj Þ as a subalgebra of DN in an obvious way.
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Proof. Let jAJ: By Lemma 2.1, we need only to consider the case that the dimension of Hj is greater than one. Assume that u; vAHj are orthogonal unit vectors. By Lemma 2.1, there exist orthogonal unit vectors x and y such that Fðu#uÞ ¼ x#x and Fðv#vÞ ¼ y#y: For any lAC; since u þ lvAHj ; we have Fððu þ lvÞ#ðu þ lvÞÞ ¼ yl #yl ; where yl AKk for some kAK and jjyl jj2 ¼ jlj2 þ 1: If dim K ¼ 2; then yl is a linear combination of the vectors x and y; and hence, both x and y are in Kk : Assume that dim K42: For any nonzero vector z lying in some atom of M orthogonal to both x and y; by Lemma 2.1, there is a nonzero vector w belonging to some atom of N orthogonal to both u and v such that Fðw#wÞ ¼ z#z: It follows from the orthogonality of w and u þ lv that z>yl : This entails again that yl is a linear combination of x and y: Hence x and y must belong to the same atom Kk of M: That is, F maps orthogonal projections of rank one in BðHj Þ into those of BðKk Þ: Denote such k ¼ yð jÞ: Because F1 also preserves the closure of numerical range, y must be a one to one map from J onto K: By identifying BðHj Þ as a subalgebra of DN in an obvious way, we claim further that FðBðHj ÞÞDBðKyð jÞ Þ: Let A be any positive operator in BðHj Þ; then FðAÞX0 and hence FðAÞADM : If FðAÞeBðKyð jÞ Þ; then there is a j1 AJ with j1 aj such that Pyð j1 Þ FðAÞPyð j1 Þ a0; where Pyð jÞ denotes the projection from K onto Kyð jÞ : Thus, there exists a unit vector xAKyð j1 Þ and a positive number r such that Pyð j1 Þ FðAÞPyð j1 Þ Xrx#x: By Lemma 2.1, we can pick uAHj1 so that Fðu#uÞ ¼ x#x: Then W ðA ru#uÞ-ðN; 0Þa| since AABðHj Þ and uAHj1 : However, W ðFðA ru#uÞÞC½0; þNÞ by virtue of FðAÞ rx#xX0; which contradicts the assumption that F preserves the closure of numerical range. So FðAÞABðKyð jÞ Þ: By noticing that F is linear when restricted to the subspace spanned by a rank-1 projection (see Lemma 2.1), a similar argument as above shows that FðiAÞABðKyð jÞ Þ: Therefore, by the real linearity of F; for every operator T in BðHj Þ; we have FðTÞABðKyð jÞ Þ: That is, FðBðHi ÞÞDBðKyð jÞ Þ: Similarly, F1 ðBðKyð jÞ ÞÞDBðHj Þ: Hence for every jAJ; FðBðHj ÞÞ ¼ BðKyð jÞ Þ: & Note that, if L is a closed subspace of H spanned by some atoms of the nest N; then for every TADN ; we have PL T ¼ TPL ; i.e., L is a reductive subspace of T: Theorem 2.3. Let N and M be two atomic nests on H and K; respectively, and let F : DN -DM be a surjective additive map. Then F preserves the closure of numerical range if and only if there exist space decompositions H ¼ X1 "X2 and K ¼ Y1 "Y2 with Xs and Ys ðs ¼ 1; 2Þ spanned by some atoms of N and M; respectively, and there exists a unitary operator U1 : X1 -Y1 and a conjugate unitary operator U2 : X2 -Y2 such that FðTÞ ¼ U1 T1 U1 "U2 T2 U2 for every T ¼ T1 "T2 ADN with Ts ¼ PXs T ðs ¼ 1; 2Þ:
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Proof. The sufficiency is clear. Now we prove the necessity. Firstly notice that F : DN -DM is a bijective additive map since F preserves the closure of numerical range. By Lemma 2.2, there exists a 1-1 and onto map y : J-K such that for every jAJ; the restriction Fj of F to BðHj Þ is a bijective additive map preserving the closure of numerical range from BðHj Þ onto BðKyð jÞ Þ: It is implied by the proof of [1, Theorem 1] that for every jAJ; either there exists a unitary operator Uj : Hj -Kyð jÞ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þ; or there exists a conjugate unitary operator Uj : Hj -Kyð jÞ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þ: Let J1 ¼ fjAJ j there exists a unitary operator Uj ABðHj ; Kyð jÞ Þ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þg; J2 ¼ fjAJ j there exists a conjugate unitary operator Uj : Hj -Kyð jÞ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þg; then J ¼ J1 ,J2 : Note that J1 or J2 may be an empty set. There is no harm in assuming that neither J1 nor J2 is empty. Put Hs ¼ "jAJs Hj and Ks ¼ "jAJs Kyð jÞ ðs ¼ 1; 2Þ; then U1 ¼ "jAJ1 Uj ABðH1 ; K1 Þ is unitary and U2 ¼ "jAJ2 Uj : H2 -K2 is conjugate unitary. Write F ¼ F1 "F2 ; where Fs ¼ Fj"jAJ BðHj Þ ðs ¼ 1; 2Þ: Set C1 ðAÞ ¼ U1 F1 ðAÞU1 for every AA"jAJ1 BðHj Þ; s then C1 ðTj Þ ¼ Tj ð8Tj ABðHj Þ; jAJ1 Þ: For each jAJ1 ; let Pj ABðHj Þ be a projection. Claim 1. If AA"jAJ1 BðHj Þ and if either AX0 or iAX0; then Pj A ¼ 0 implies Pj C1 ðAÞ ¼ C1 ðAÞPj ¼ 0: It is clear that we also have APj ¼ 0: For a sufficiently large positive number M; wðA þ MPj Þ ¼ M: It follows that wðC1 ðAÞ þ MPj Þ ¼ M: Hence for any unit vector xArngPj (the range of Pj ), j/C1 ðAÞx; xS þ MjpM:
ð2:1Þ
If AX0; then /C1 ðAÞx; xSX0; therefore inequality (2.1) entails /C1 ðAÞx; xS ¼ 0: and j/C1 ðAÞx; xS þ Mj2 ¼ If iAX0; then Reð/C1 ðAÞx; xSÞ ¼ 0 j/C1 ðAÞx; xSj2 þ M 2 ; where ReðrÞ denotes the real part of complex number r: By (2.1) again we get /C1 ðAÞx; xS ¼ 0: Thus Pj C1 ðAÞPj ¼ 0: Since C1 ðAÞX0 or iC1 ðAÞX0; one must have Pj C1 ðAÞ ¼ C1 ðAÞPj ¼ 0; as desired. Claim 2. For any projection PA"jAJ1 BðHj Þ; C1 ðPÞ ¼ P and C1 ðiPÞ ¼ iP: For any projection PA"jAJ1 BðHj Þ; P can be written into P ¼ "jAJ1 Pj with Pj being projection in BðHj Þ: Since Pj ðP Pj Þ ¼ ðP Pj ÞPj ¼ 0; by Claim 1,
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Pj ðC1 ðPÞ Pj Þ ¼ ðC1 ðPÞ Pj ÞPj ¼ 0; so C1 ðPÞPj ¼ Pj C1 ðPÞ ¼ Pj ; and therefore, PC1 ðPÞ ¼ C1 ðPÞP ¼ P: Hence, to prove C1 ðPÞ ¼ P; one need only to show that, for every jAJ1 and every xAHj ; Px ¼ 0 will imply that C1 ðPÞx ¼ 0 since N is atomic. Assume that xAHj for some jAJ1 such that Px ¼ 0; then Pðx#xÞ ¼ 0: By Claim 1 again, we get C1 ðPÞðx#xÞ ¼ 0; which implies that C1 ðPÞx ¼ 0: So C1 ðPÞ ¼ P: Similarly, one can prove C1 ðiPÞ ¼ iP ð8PA"jAJ1 BðHj ÞÞ by using Claim 1 and the fact that C1 ðiPj Þ ¼ iPj : Hence Claim 2 is true. Since "jAJ1 BðHj Þ is a von Neumann algebra and C1 is continuous in norm topology, we see that C1 ðAÞ ¼ A for all AA"jAJ1 BðHj Þ: So F1 ðAÞ ¼ U1 AU1 for all AA"jAJ1 BðHj Þ: A similar argument leads to F2 ðBÞ ¼ U2 B U2 for all BA"jAJ2 BðHj Þ: Now for every TADN ; write T ¼ T1 "T2 with Ts A"jAJs BðHj Þ ðs ¼ 1; 2Þ; we have FðTÞ ¼ U1 T1 U1 "U2 T2 U2 : The proof is completed. & Theorem 2.3 can also be restated as follows, which is sometimes convenient for applications. Theorem 2:30 . Let N and M be two atomic nests on H and K; respectively, and let F : DN -DM be a surjective additive map. Then F preserves the closure of numerical range if and only if there exist space decompositions H ¼ X1 "X2 and K ¼ Y1 "Y2 with Xs and Ys spanned by some atoms of N and M; respectively, and there exist unitary operators Us : Xs -Ys ðs ¼ 1; 2Þ such that FðTÞ ¼ U1 T1 U1 "U2 T2tr U2 for every T ¼ T1 "T2 ADN with Ts ¼ PXs T ðs ¼ 1; 2Þ; where T2tr denotes the transpose of T2 relative to a base of X2 which consists of arbitrarily fixed bases of the atoms of N contained in X2 : Remark 2.1. Let N; M and F : Alg N-Alg M be just as assumed in Lemma 2.2. Then, by Lemma 2.2 and the proof of Theorem 2.3, FjDN is a linear bijective map from DN onto DM and FjBðHj Þ has the form Ui ðÞUi or Ui ðÞ Ui ; where Ui is unitary in the former case and conjugate unitary in the latter case.
3. Additive maps preserving the closure of numerical range on maximal atomic nest algebras In this section we consider the additive maps preserving the closure of numerical range between nest algebras with maximal atomic nests. We first give a basic lemma on general atomic nest algebras which is useful in both this section and the next section. Lemma 3.1. Let N and M be two atomic nests on complex Hilbert spaces H and K; respectively, and let fHj j jAJg and fKk j kAKg be the sets of all atoms of N and M; respectively. Assume F : Alg N-Alg M is an additive surjective map and preserves
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the closure of numerical range. Then there exists a 1-1 and onto map y : J-K such that the following statements are true. (1) For any i; jAJ with ioj; and for any Tij APi Alg NPj ; PyðiÞ FðTij ÞPyð jÞ ¼ FðTij Þ if yðiÞoyð jÞ or Pyð jÞ FðTij ÞPyðiÞ ¼ FðTij Þ if yð jÞoyðiÞ: (2) For any unit vectors xi AHi and yj AHj with ioj; there exist unit vectors uyðiÞ AKyðiÞ ; vyð jÞ AKyð jÞ and complex unit aij AC such that Fðxi #xi Þ ¼ uyðiÞ #uyðiÞ ; Fðyj #yj Þ ¼ vyð jÞ #vyð jÞ and Fðxi #yj Þ ¼ aij uyðiÞ #vyð jÞ if yðiÞo yð jÞ or Fðxi #yj Þ ¼ aij vyð jÞ #uyðiÞ if yð jÞoyðiÞ: Proof. Take y : J-K just as in Lemma 2.2. (1) We claim that, for pairwise different i; j; lAJ with ioj and for any Tij APi Alg NPj ; PyðlÞ FðTij Þ ¼ 0 and FðTij ÞPyðlÞ ¼ 0; where PyðlÞ denotes the projection from K onto KyðlÞ : Assume that there exists a unit vector uAKyðlÞ such that /FðTij Þu; uSa0: By Lemma 2.1, we can find a unit vector zAHl so that Fðz#zÞ ¼ u#u: Take lAC such that jlj4jjTij jj and jl þ /FðTij Þu; uSj ¼ jlj þ j/FðTij Þu; uSj: By Lemma 2.1, Fðlz#zÞ ¼ lu#u: So wðlz#z þ Tij Þ ¼ jljojlj þ j/FðTij Þu; uSjpwðlu#u þ FðTij ÞÞ; which contradicts the assumption that F preserves the closure of numerical range. Hence /FðTij Þu; uS ¼ 0 for all uAKyðlÞ ; which implies that /FðTij Þu; vS ¼ 0 for all u; vAKyðlÞ since KyðlÞ is a complex Hilbert space. Therefore PyðlÞ FðTij ÞPyðlÞ ¼ 0 for all Tij APi Alg NPj : If there exists some Tij APi Alg NPj such that PyðlÞ FðTij Þa0; then there exists a unit vector uAKyðlÞ such that /FðTij Þv; uSa0 for some vAKyðhÞ ðhAJÞ: It is clear that loh: Take unit vector zAHl such that Fðz#zÞ ¼ u#u and take a complex number l such that jlj4jjTij jj: Then Fðlz#z þ Tij Þ has a principal submatrix of the l /FðTij Þv; uS ; where gAC: Therefore, form B ¼ 0 g wðlz#z þ Tij Þ ¼ jljowðBÞpwðFðlz#z þ Tij ÞÞ; a contradiction. Hence PyðlÞ FðTij Þ ¼ 0: FðTij ÞPyðlÞ ¼ 0 can be proved similarly. So the claim is true. For any Tij APi Alg NPj ; let FðTij Þ ¼ A: Without loss of generality, we assume that yðiÞoyð jÞ: Suppose that there exists a unit vector uAKyðiÞ such that /Au; uSa0: Then there is a unit vector zAHi such that Fðz#zÞ ¼ u#u: For any lAC satisfying jl þ /Au; uSj ¼ jlj þ j/Au; uSj; let S ¼ lz#z þ Tij : Then SS ¼ jlj2 z#z þ Tij Tij ; and hence, wðSÞ2 pjjSjj2 pjlj2 þ jjTij jj2 : Since F preserves the closure of numerical range, we have ðjlj þ j/Au; uSjÞ2 pwðlu#u þ AÞ2 pjlj2 þ jjTij jj2 ; this would lead to a contradiction that 2j/Au; uSjjlj þ j/Au; uSj2 pjjTij jj2
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for all positive numbers jlj: Therefore, we must have /Au; uS ¼ 0 for all uAKyðiÞ ; and consequently, PyðiÞ FðTij ÞPyðiÞ ¼ 0: Similarly, Pyð jÞ FðTij ÞPyð jÞ ¼ 0: Thus we have proved that PyðiÞ FðTij ÞPyð jÞ ¼ FðTij Þ: (2) For any unit vectors xi AHi and yj AHj ; let Fðxi #yj Þ ¼ A: Without loss of generality, we assume that yðiÞoyð jÞ: By (1), PyðiÞ Fðxi #yj ÞPyð jÞ ¼ Fðxi #yj Þ: Let Fðxi #xi Þ ¼ uyðiÞ #uyðiÞ and Fðyj #yj Þ ¼ vyð jÞ #vyð jÞ ; then uyðiÞ AKyðiÞ and vyð jÞ AKyð jÞ are unit vectors. Pick unit vectors uAKyðiÞ and vAKyð jÞ such that /Av; uSa0: Correspondingly, there exist unit vectors xAHi and yAHj such that Fðx#xÞ ¼ u#u and Fðy#yÞ ¼ v#v: Assume that u>uyðiÞ ; then x>xi : Take lAC with jlj41: we see that Fðlx#x þ xi #yj Þ has a principal submatrix l /Av; uS ; whose numerical radius is larger than jlj ¼ wðlx#x þ xi #yj Þ; 0 0 this is impossible. Assume that v>vyð jÞ ; then y>yj and Fðly#y þ xi #yj Þ contains a 0 /Av; uS principal submatrix ; which has the numerical radius larger than 0 l jlj ¼ wðlx#x þ xi #yj Þ; again a contradiction. Therefore, A ¼ Fðxi #yj Þ is linearly dependent to uyðiÞ #vyð jÞ : Since F preserves the closure of numerical range, there exists aij AC with jaij j ¼ 1 such that Fðxi #yj Þ ¼ aij uyðiÞ #vyð jÞ : & In the rest of this section, we always assume that N and M are two maximal atomic nests on Hilbert spaces H and K; and fHi j iAJg and fKk j kAKg be the sets of all atoms of N and M; respectively. We also suppose in the sequel that F : Alg N-Alg M is an additive surjective map which preserves the closure of numerical range and y : J-K is the 1-1 and onto map stated in Lemma 3.1. Note that F is real linear. For each iAJ; pick out a unit vector ei AHi ; then fei j iAJg is an orthonormal base of H since N is maximal atomic. Thus, for i; jAJ; ioj if and only if iaj and ei #ej AAlg N: Since M is maximal atomic, we can also choose unit vectors fk AKk such that ffk j kAKg is an orthonormal base of K: Lemma 3.2. For any j; kAJ with jok and for any xAC; Fðxej #ek Þ ¼ dðxÞFðej #ek Þ; where dðxÞ ¼ x for every xAC or dðxÞ ¼ x% for every xAC: Proof. For jAJ; replacing ej by iej in fek j kAJg; then fiej ; ek j kAJ and kajg is still a base of H: By Lemma 3.1, for j; kAJ and jok; we have Fðð1 þ iÞej #ek Þ ¼ ajk fyð jÞ #fyðkÞ þ lfyð jÞ #fyðkÞ if yð jÞoyðkÞ or Fðð1 þ iÞej #ek Þ ¼ ajk fyðkÞ #fyð jÞ þ lfyðkÞ #fyð jÞ if yðkÞoyð jÞ; where jlj ¼ 1: Since F preserves the numerical radius, pffiffiffi we must have jl þ ajk j ¼ 2; and hence l ¼ 7iajk : Now the existence of d is obvious from the real linearity of F; completing the proof. & Lemma 3.3. Assume that y : J-K is an order isomorphism (or, an anti-order isomorphism). For any i; k; jAJ with iokoj; let aij ; aik and akj be as in Lemma 3.1, then aij ¼ aik akj :
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Proof. Suppose that y : J-K is an order isomorphism. Let A ¼ ei #ek þ ek #ej þ ei #ej ; then, by Lemma 3.1, FðAÞ ¼ aik fyðiÞ #fyðkÞ þ akj fyðkÞ #fyð jÞ þ aij fyðiÞ #fyð jÞ with jaik j ¼ jakj j ¼ jaij j ¼ 1: One can check that A þ A has the smallest eigenvalue 1 and the largest eigenvalue 2: Since W ðAÞ ¼ W ðFðAÞÞ; FðAÞ þ FðAÞ also has 1 and 2 as the smallest and the largest eigenvalue, respectively. Let L be the subspace spanned by ffyðkÞ ; fyðiÞ ; fyð jÞ g; then it follows from detððFðAÞ þ FðAÞ þ IÞjL Þ ¼ 0 and jaik akj aij j ¼ 1 that aik akj aij ¼ 1; where det B denotes the determinant of matrix B: Hence aij ¼ aik akj : The proof is similar for the case that y : J-K is an anti-order isomorphism. & Lemma 3.4. Assume that y : J-K is an order isomorphism (or, an anti-order isomorphism). Then for all j; kAJ with jok; Fðiej #ek Þ ¼ iFðej #ek Þ: Proof. Let jokol and A ¼ sej #ek þ tej #el þ rek #el ; where s; t; rAC: Assume that y : J-K is an order isomorphism. By Lemmas 3.1 and 3.2, FðAÞ ¼ d1 ðsÞajk fyð jÞ #fyðkÞ þ d2 ðtÞajl fyð jÞ #fyðlÞ þ d3 ðrÞakl fyðkÞ #fyðlÞ ; here dm ðxÞ ¼ x ð8xACÞ or dm ðxÞ ¼ x% ð8xACÞ; m ¼ 1; 2; 3: This gives eight possible expressions of FðAÞ according to the following cases: Case 1: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs; t; rÞ or ðs%; t%; r%Þ: Case 2: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs%; t; rÞ or ðs; t%; r%Þ: Case 3: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs; t; r%Þ or ðs%; t%; rÞ: Case 4: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs; t%; rÞ or ðs%; t; r%Þ: Let L1 ¼ spanfej ; ek ; el g and L2 ¼ spanffyð jÞ ; fyðkÞ ; fyðlÞ g: Since F preserves the closure of numerical range, A þ A and FðAÞ þ FðAÞ have the same largest and smallest eigenvalues. Note that, since the coefficients of l2 are zero in both detððA þ A lÞjL1 Þ and detððFðAÞ þ FðAÞ lÞjL2 Þ; the largest and smallest roots of these two polynomials determine their third root. It turns out that detððA þ A lÞjL1 Þ ¼ detððFðAÞ þ FðAÞ lÞjL2 Þ: That is, l3 þ ðjsj2 þ jtj2 þ jrj2 Þl þ 2 Reðst%rÞ ¼ l3 þ ðjsj2 þ jtj2 þ jrj2 Þl þ 2 Reðd1 ðsÞd2 ðtÞd3 ðrÞÞ: It is clear now that Cases 1–3 cannot occur. For Case 4, i.e., d1 ðsÞ ¼ s; d2 ðtÞ ¼ t and d3 ðrÞ ¼ r or d1 ðsÞ ¼ s%; d2 ðtÞ ¼ t% and d3 ðrÞ ¼ r%; we claim that dm ðxÞ ¼ x% ðm ¼ 1; 2; 3Þ is not possible. Taking s ¼ i þ 2; t ¼ 1 i and r ¼ 1 þ i; it can be checked that the largest or smallest eigenvalues of AA and FðAÞFðAÞ are different, a contradiction. So i i dm ðxÞ ¼ x ðm ¼ 1; 2; 3Þ; this shows that, for all j; kAJ with jok; Fðiej #ek Þ ¼ iFðej #ek Þ: The case that y : J-K is an anti-order isomorphism can be treated with similarly. &
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Lemma 3.5. ð1Þ If y : J-K is an order isomorphism, then there exists a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M such that FðF Þ ¼ UFU for every F AAlgF N: ð2Þ If y : J-K is an anti-order isomorphism, then there exists a conjugate unitary operator U from H onto K satisfying UðN> Þ ¼ M such that FðF Þ ¼ UF U for every F AAlgF N: Proof. Fix some i0 AJ: Let UABðH; KÞ be defined by ( Uei ¼
aii0 fyðiÞ
if ipi0 ;
ai0 i fyðiÞ
if i4i0 :
It is clear that U is unitary. If y : J-K is an order isomorphism, then UðNÞ ¼ M: For every ei #ej AAlg N; since
Fðei #ej Þ ¼ aij fyðiÞ #fyð jÞ
8 > < aii0 aji0 fyðiÞ #fyð jÞ ¼ aii0 ai0 j fyðiÞ #fyð jÞ > : ai0 i ai0 j fyðiÞ #fyð jÞ
if ipjpi0 ; if ipi0 pj; if i0 pipj;
we have Fðei #ej Þ ¼ Uei #Uej : For any rank-1 operator x#yAAlgF N; there exists P P" i1 such that x ¼ " ipi1 xi ei and y ¼ jXi1 Zj ej ; so " X
x#y ¼
ipi1
! xi e i #
" X
! Zj e j
jXi1
¼
" X " X
xi Zj ei #ej :
ipi1 jXi1
The boundedness of F; together with Lemma 3.4, implies that X X Fðx#yÞ ¼ xi Zj Fðei #ej Þ ¼ xi Zj Uei #Uej ¼ Uðx#yÞU : ipi1 pj
ipi1 pj
So FðF Þ ¼ UFU for every F AAlgF N; i.e., (1) holds. If y : J-K is an anti-order isomorphism, then it is clear that UðN> Þ ¼ M: By Lemma 3.3, we have Fðei #ej Þ ¼ aij fyð jÞ #fyðiÞ ¼ Uej #Uei : It follows that Fðx#yÞ ¼
X ipi1 pj
xi Zj Fðei #ej Þ ¼
X
xi Zj Uej #Uei ¼ Uðx#yÞ U :
ipi1 pj
So FðF Þ ¼ UF U for all F AAlgF N; i.e., (2) holds.
&
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Let LCJ be a subset, then L is a totally ordered subset inheriting the order of J: Set PL be the projection from H onto spanfel j lALg (the closed linear span of fel j lALg). Lemma 3.6. Let LCJ be a subset and FL ¼ FjPL Alg NPL : Then FL : PL Alg NPL -PyðLÞ Alg MPyðLÞ is an additive bijective map which preserves the closure of numerical range. Proof. We first show that FL ðPL Alg NPL ÞDPyðLÞ Alg MPyðLÞ : Assume that there exists F APL Alg NPL such that FðF ÞePyðLÞ Alg MPyðLÞ ; then there is ieL or jeL such that PyðiÞ FðF ÞPyð jÞ ¼ mfyðiÞ #fyð jÞ a0 (assume that yðiÞpyð jÞ). We might as well assume that ipj: Let T ¼ aei #ei þ F ; where aAC and jaj4jjF jj: It is clear that wðTÞ ¼ jaj: Write PyðiÞ FðTÞPyðiÞ ¼ bfyðiÞ #fyðiÞ : We may assume that a is also taken so that jaþ bj ¼ jaj þ jbj; then FðTÞ has a principal matrix of the from aþb m ; where gAC; so wðFðTÞÞ4ja þ bjXjaj ¼ wðTÞ; a contradiction. Hence 0 g FL ðPL Alg NPL ÞDPyðLÞ Alg MPyðLÞ : The above argument applied to F1 yields F1 ðPyðLÞ AlgMPyðLÞ ÞDPL AlgNPL : Therefore, FL ðPL AlgNPL Þ ¼ PyðLÞ Alg MPyðLÞ : For any F APL Alg NPL ; we have F ¼ PL FPL : Let T ¼ F þ sðI PL Þ; where sAC: Then, by Remark 2.1, FðTÞ ¼ FL ðF Þ þ sðI PyðLÞ Þ: So convfW ðFL ðF ÞÞ,fsgg ¼ W ðFðTÞÞ ¼ W ðTÞ ¼ convfW ðF Þ,fsgg: Taking sAW ðF Þ; one gets W ðFL ðF ÞÞDW ðF Þ; picking sAW ðFL ðF ÞÞ; one obtains W ðF ÞDW ðFL ðF ÞÞ: So for every F APL Alg NPL ; we always have W ðFL ðF ÞÞ ¼ W ðF Þ: & Let Eij denote the standard matrix units of Mn ðCÞ ði; j ¼ 1; y; nÞ and Tn be the upper triangular matrix algebras in Mn ðCÞ: The proof of the following lemma can be found in [11]. Lemma 3.7. Let f : Tn -Tn be an additive map which preserves the numerical range. Then there exists a 1-1 map W : f1; 2; y; ng-f1; 2; y; ng which satisfies fðEii Þ ¼ EWðiÞWðiÞ ð1pipnÞ; and there exists jAf1; 2; y; ng such that Wð jÞ ¼ 1 and one of the following holds: (1) Wð jÞoWð j þ 1Þo?oWðn 1ÞoWðnÞoWð1ÞoWð2Þo?oWð j 1Þ: (2) Wð jÞoWð j 1Þo?oWð2ÞoWð1ÞoWðnÞoWðn 1Þo?oWð j þ 1Þ: Let Js ðs ¼ 1; 2Þ be totally ordered sets, we denote the order sum of J1 and J2 by J ¼ J1 þ J2 ; which is a totally ordered set J ¼ J1 ,J2 (regard as J1 -J2 ¼ |), in
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which ioj means ioj if i; jAJs ; or if iAJ1 and jAJ2 : Note that, by the definition, J1 þ J2 aJ2 þ J1 in general. Lemma 3.8. isomorphism, ð1Þ There y : Js -yðJs Þ ð2Þ There y : Js -yðJs Þ
If y : J-K is neither an order isomorphism nor an anti-order then one of the following holds: exist J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ1 Þ þ yðJ2 Þ and is an anti-order isomorphism, s ¼ 1; 2: exist J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ2 Þ þ yðJ1 Þ and is an order isomorphism, s ¼ 1; 2:
Proof. Assume that y : J-K is neither an order isomorphism nor an anti-order isomorphism, then there exist i1 ; i2 ; i3 AJ with i1 oi2 oi3 such that one of the following holds: (i) (ii) (iii) (iv)
yði1 Þoyði3 Þoyði2 Þ; yði2 Þoyði3 Þoyði1 Þ; yði2 Þoyði1 Þoyði3 Þ; yði3 Þoyði1 Þoyði2 Þ:
Suppose that (i) holds. For any j1 ; j2 ; j3 AJ with j1 oj2 oj3 ; it is easily checked that fi1 ; i2 ; i3 ; j1 ; j2 ; j3 g has totally the following 20 possible orders: ð11Þ ð21Þ ð31Þ ð41Þ ð51Þ ð61Þ ð71Þ ð81Þ ð91Þ ð101Þ ð111Þ ð121Þ ð131Þ ð141Þ ð151Þ ð161Þ ð171Þ ð181Þ ð191Þ ð201Þ
i1 oi2 oi3 pj1 oj2 oj3 ; i1 oi2 pj1 oi3 pj2 oj3 ; i1 pj1 oi2 oi3 pj2 oj3 ; i1 oi2 pj1 oj2 pi3 oj3 ; i1 pj1 oj2 pi2 oi3 pj3 ; i1 pj1 oi2 pj2 oi3 pj3 ; i1 pj1 oj2 oj3 pi2 oi3 ; i1 pj1 oj2 pi2 oj3 pi3 ; i1 pj1 oi2 pj2 oj3 pi3 ; i1 oi2 pj1 oj2 oj3 pi3 ; j1 pi1 oi2 pj2 oj3 pi3 ; j1 oj2 pi1 oi2 pj3 oi3 ; j1 oj2 pi1 oj3 pi2 oi3 ; j1 oj2 oj3 pi1 oi2 oi3 ; j1 pi1 oj2 pi2 oj3 pi3 ; j1 pi1 oj2 oj3 pi2 oi3 ; j1 pi1 oi2 oi3 pj2 oj3 ; j1 pi1 oi2 pj2 oi3 pj3 ; j1 pi1 oj2 pi2 oi3 pj3 ; j1 oj2 pi1 oi2 oi3 pj3 :
Let L1 ¼ fi1 ; i2 ; i3 ; j1 ; j2 ; j3 g and L2 ¼ yðL1 Þ: By Lemma 3.6, FjPL
1
Alg NPL1
: PL1 Alg NPL1 -PL2 Alg NPL2
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preserves the numerical range. Using Lemma 3.7, it is easily checked that, in each of 20 possible cases listed above, fyð j1 Þ; yð j2 Þ; yð j3 Þg can only have one of the order relations (i), (iii) and the anti-order yð j3 Þoyð j2 Þoyð j1 Þ: Similarly, that (iii) holds also implies that fyð j1 Þ; yð j2 Þ; yð j3 Þg can only have one of the order relations (i), (iii) and the anti-order. Now we consider the following three cases. Case 1: (i) occurs but (iii) does not. Take i0 such that i0 oi1 oi2 ; then yði0 Þoyði2 Þoyði1 Þ; which contradicts yði1 Þoyði2 Þ: So there is no i0 such that i0 oi1 : Hence, i1 ¼ min J and yði1 Þ ¼ min K: Therefore, for any j2 ; j3 ; y; jn with i1 oj2 oj3 o?ojn ; we have yði1 Þoyð jn Þoyð jn1 Þo?oyð j2 Þ: Put J ¼ J1 þ J2 and K ¼ K1 þ K2 ; where J1 ¼ fi1 g and K1 ¼ fyði1 Þg; then y : J2 -K2 is an anti-order isomorphism. Case 2: (iii) occurs but (i) does not. A similar argument just as in Case 1 implies that there exists j1 AJ such that j1 ¼ max J and yð j1 Þ ¼ max K: Let J ¼ J1 þ J2 and K ¼ K1 þ K2 ; where J2 ¼ fj1 g and K2 ¼ fyð j1 Þg; we have y : J1 -K1 is an anti-order isomorphism. Case 3: Both (i) and (iii) occur. Set J1 ¼ fiAJ j there are j; k such that iojok and yðiÞoyðkÞoyð jÞg; J2 ¼ flAJ j there are j; k such that jokol and yðkÞoyð jÞoyðlÞg: Note that iAJ1 and hoi imply hAJ1 ; lAJ2 and lom imply mAJ2 : Notice also that, if h; iAJ1 and hoi; then yðhÞ4yðiÞ; if l; mAJ2 and lom; then yðlÞ4yðmÞ: By Cases 1 and 2, J1 a| and J2 a|: If iAJ1 -J2 ; then there are gohoiokol such that yðhÞoyðgÞoyðiÞoyðlÞoyðkÞ; this is impossible by Lemmas 3.6 and 3.7. So J1 -J2 ¼ |; and therefore, J1 oJ2 ; i.e., for any iAJ1 and lAJ2 ; we have iol: We assert that for any iAJ1 and lAJ2 ; yðiÞoyðlÞ: Otherwise, assume that yðlÞoyðiÞ: Take j; kAJ such that jokol; then yðkÞoyð jÞoyðlÞ: It is clear that fi; j; k; lg have three possible order relations: ð10 Þ ipjokol; ð20 Þ joipkol; ð30 Þ jokoiol: However, by Lemmas 3.6 and 3.7, ð10 Þ would imply yðkÞoyð jÞpyðiÞoyðlÞoyðiÞ and ð20 Þ would imply yðkÞpyðiÞoyð jÞoyðlÞoyðiÞ; contradictions. ð30 Þ would imply that yðiÞoyðkÞoyð jÞoyðlÞoyðiÞ or yðkÞoyð jÞoyðlÞoyðiÞ: The former is impossible and for the latter, we would have iAJ2 ; a contradiction. So the assertion holds. If J ¼ J1 ,J2 ; then the proof is completed. Assume that JaJ1 ,J2 ; then J\ðJ1 ,J2 Þa|: For any jAJ\ðJ1 ,J2 Þ; we have J1 ojoJ2 : Since H is infinite dimensional, there is at least one among J1 ; J3 ¼ J\ðJ1 ,J2 Þ and J2 which is an % ¼ J\ðJ ,J ÞX3; where S % denotes the cardinal of the set infinite set. Note that if J 3 1 2 S; we may take i; j; kAJ3 such that iokoj: Let hAJ1 ; lAJ2 ; then hoiojokol: Since yðhÞoyðlÞ; we have yðkÞoyð jÞoyðiÞoyðhÞoyðlÞ or yðhÞoyðlÞ oyðkÞoyð jÞoyðiÞ; so y is anti-order when restricted to J3 and either
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yðJ3 ÞoyðJ1 Þ or yðJ2 ÞoyðJ3 Þ: If yðJ3 ÞoyðJ1 Þ occurs, let J01 ¼ J1 þ J3 ; J02 ¼ J2 ; if yðJ2 ÞoyðJ3 Þ occurs, let J01 ¼ J1 ; J02 ¼ J3 þ J2 : Then J ¼ J01 þ J02 ; yðJ01 ÞoyðJ02 Þ and y is anti-order on J01 and J02 : If 1pJ% 3 p2; then at least one of J1 and J2 is an infinite set. Without loss of generality, we assume that J1 is an infinite set. For any jAJ3 ; pick i1 ; i2 AJ1 with i1 oi2 and lAJ2 ; then i1 oi2 ojol: Due to yðis ÞoyðlÞ ðs ¼ 1; 2Þ; we have either (a) yði2 Þoyði1 ÞoyðlÞoyð jÞ; or (b) yð jÞoyði2 Þoyði1 ÞoyðlÞ: We claim that (a) and (b) cannot occur simultaneously. Indeed, if (a) holds for i1 oi2 ojol; and (b) holds for h1 oh2 ojom; then yði2 Þoyði1 ÞoyðlÞo yð jÞoyði2 Þoyði1 ÞoyðlÞ; and therefore, i1 oi2 omoh1 oh2 ojol or h1 oh2 ojoloi1 oi2 om; which contradicts to the assumption h1 oh2 ojom or i1 oi2 ojol: Hence yðJ3 ÞoyðJ1 Þ or yðJ3 Þ4yðJ2 Þ: If J3 ¼ fj; kg with jok; then it can be seen from yðJ3 Þ4yðJ2 Þ or yðJ3 ÞoyðJ1 Þ that yðkÞoyð jÞ: So y is still antiorder on J3 if 1pJ% 3 p2: In the case yðJ3 ÞoyðJ1 Þ; let J01 ¼ J1 þ J3 ; J02 ¼ J2 ; in the case yðJ3 Þ4yðJ2 Þ; let J01 ¼ J1 ; J02 ¼ J3 þ J2 ; then J ¼ J01 þ J02 ; as desired. All in all, if (i) or (iii) occurs, then there exists a decomposition J ¼ J1 þ J2 such that yðJ1 ÞoyðJ2 Þ and, when restricted to Js ; y : Js -yðJs Þ is an anti-order isomorphism ðs ¼ 1; 2Þ: If (ii) or (iv) occurs, one can similarly prove that there exists a decomposition J ¼ J1 þ J2 such that yðJ1 Þ4yðJ2 Þ and y : Js -yðJs Þ is an order isomorphism (s ¼ 1; 2). & The following examples show that there are many additive maps on nest algebras which preserve numerical ranges but are neither linear nor conjugate linear. Let Ns be a nest on Hilbert space Hs ; s ¼ 1; 2; and let N ¼ N1 þ N2 consist of N1 "0 and H1 "N2 ; where Ns ANs ðs ¼ 1; 2Þ: It is clear that N is a nest on H ¼ H1 "H2 : Example 3.1. Let N ¼ N1 þ N2 and M ¼ M2 þ M1 be two nests on Hilbert spaces H ¼ H1 "H2 and K ¼ K2 "K1 ; respectively. Assume that unitary operators Us ABðHs ; Ks Þ satisfy Us ðNs Þ ¼ Ms ðs¼ 1; 2Þ: For TAAlg N; with respect to the T1 T12 above decompositions, one has T ¼ with Ts AAlg Ns ðs ¼ 1; 2Þ and 0 T2 U2 T2 U2 U2 T12 U1 T12 ABðH2 ; H1 Þ: Define G : Alg N-Alg M by GðTÞ ¼ : It 0 U1 T1 U1 is clear GðTÞAAlg M: Let x ¼ x1 "x2 AH with jjxjj ¼ 1 and let y ¼ aU2 x2 "U1 x1 ; where ( a¼
1
if /T12 x2 ; x1 S ¼ 0;
/x1 ;T12 x2 S /T12 x2 ;x1 S
if /T12 x2 ; x1 Sa0:
ð3:1Þ
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Then /GðTÞy; yS ¼ /aU2 T2 x2 ; aU2 x2 S þ /U1 T1 x1 ; U1 T1 x1 S þ /U2 T12 x1 ; aU2 x2 S
¼ /T2 x2 ; x2 S þ /T1 x1 ; x1 S þ /T12 x2 ; x1 S ¼ /Tx; xS: We conclude that W ðGðTÞÞ+W ðTÞ: On the other hand, for any unit vector y ¼ y2 "y1 AK ¼ K2 "K1 ; let x ¼ bU1 y1 "U2 y2 AH ¼ H1 "H2 ; where ( b¼
1
if /T12 U2 y2 ; U1 y1 S ¼ 0;
/T12 U2 y2 ;U1 y1 S /U1 y1 ;T12 U2 y2 S
if /T12 U2 y2 ; U1 y1 Sa0:
ð3:2Þ
A computation shows that % /Tx; xS ¼ /T1 U1 y1 ; U1 y1 S þ /T2 U2 y2 ; U2 y2 S þ b/T 12 U2 y2 ; U1 y1 S U1 y1 ; y2 S ¼ /U1 T1 U1 y1 ; y1 S þ /U2 T2 U2 y2 ; y2 S þ /U2 T12
¼ /GðTÞy; yS: This implies that W ðGðTÞÞDW ðTÞ: So W ðGðTÞÞ ¼ W ðTÞ ð8TAAlg NÞ: Obviously, G is a surjective additive map, but G is neither linear nor conjugate linear. Example 3.2. Let N ¼ N1 þ N2 and M ¼ M1 þ M2 be two nests on Hilbert spaces H ¼ H1 "H2 and K ¼ K1 "K2 ; respectively. Assume that conjugate unitary operators Us : Hs -Ks satisfy Us ðN> For TAAlg N; s Þ ¼ Ms ðs ¼ 1; 2Þ: T1 T12 write T ¼ with Ts AAlg Ns ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: Let 0 T2 U1 T1 U1 U1 T12 U2 D : Alg N-Alg M be defined by DðTÞ ¼ : Clearly DðTÞA 0 U2 T2 U2 Alg M and D is additive and surjective but is neither linear nor conjugate linear. We claim that D preserves the numerical range. For any x ¼ x1 "x2 AH with jjxjj ¼ 1: Put y ¼ aU1 x1 "U2 x2 ; where a is taken as in (3.1). Then jjyjj ¼ 1 and /DðTÞy; yS ¼ /aU1 T1 x1 ; aU1 x1 S þ /U2 T2 x2 ; U2 x2 S þ /U1 T12 x2 ; aU1 x1 S ¼ /x1 ; T1 x1 S þ /x2 ; T2 x2 S þ /T12 x2 ; x1 S ¼ /Tx; xS: So W ðTÞDW ðDðTÞÞ: For any unit vector y ¼ y1 "y2 ; put x ¼ bU1 y1 "U2 y2 ; where b is taken as in (3.2). It is easily seen that /DðTÞy; yS ¼ /Tx; xS; and hence W ðDðTÞÞDW ðTÞ: Therefore D preserves the numerical range.
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Note that G and D in the above examples are weakly continuous. To our surprise, from the main result Theorem 4.1 in next section, for the weakly continuous and surjective additive maps between atomic nest algebras which preserve the closure of numerical range and which are neither linear nor conjugate linear, Examples 3.1 and 3.2 are only possible ones. We first prove this in the next result for the case of maximal atomic nest algebras. Let N be a nest on H: For any H1 AN; set N1 ¼ fN-H1 j NANg and N2 ¼ fN-H2 j NANg; where H2 ¼ H1> : Then Ns is a nest on Hs ðs ¼ 1; 2Þ and N ¼ N1 þ N2 : We say this space decomposition and the corresponding nest decomposition by H1 : Thus for every TAAlg N; T can be written as is determined T1 T12 T¼ ; where Ts AAlg Ns ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: 0 T2 Theorem 3.9. Let N and M be two maximal atomic nests on Hilbert spaces H and K; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective additive map. Then F preserves the closure of numerical range if and only if one of the following holds: (1) There exists a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M such that FðTÞ ¼ UTU for every TAAlg N: (2) There exists a conjugate unitary operator U from H onto K satisfying UðN> Þ ¼ M such that FðTÞ ¼ UT U for every TAAlg N: (3) There exist H1 AN; K1 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively; there exist conjugate unitary operators Us : Hs -Ks satisfying Us ðN> s Þ ¼ Ms ðs ¼ 1; 2Þ; such that ! ! T1 T12 U1 T1 U1 U1 T12 U2 ; 8T ¼ AAlg N: FðTÞ ¼ 0 U2 T2 U2 0 T2 (4) There exist H1 AN; K2 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 ; respectively; there exist unitary operators Us : Hs -Ks satisfying Us ðNs Þ ¼ Ms ðs ¼ 1; 2Þ; such that ! ! U2 T2 U2 U2 T12 U1 T1 T12 FðTÞ ¼ ; 8T ¼ AAlg N: 0 U1 T1 U1 0 T2 Proof. Let fHi j iAJg and fKj j jAKg be the set of all atoms of N and M; respectively. By Lemma 2.2, y : J-K is one to one and onto. If y : J-K is an order isomorphism or an anti-order isomorphism, then by Lemma 3.5 and the weak continuity of F; we have that the result (1) or (2) holds.
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Now suppose that y : J-K is neither an order isomorphism nor an anti-order isomorphism. By Lemma 3.8, one of the following holds: (i) There exist y : Js -yðJs Þ (ii) There exist y : Js -yðJs Þ
J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ1 Þ þ yðJ2 Þ and is an anti-order isomorphism for each s ðs ¼ 1; 2Þ: J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ2 Þ þ yðJ1 Þ and is an order isomorphism for each s ðs ¼ 1; 2Þ:
Assume that (i) holds. Let K1 ¼ yðJ1 Þ; K2 ¼ yðJ2 Þ and let H1 ¼ spanfej j jAJ1 gAN; K1 ¼ spanffk j kAK1 gAM: Then H1 and K1 determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively. It is clear that Ns and Ms are maximal atomic nests on Hs and Ks ðs ¼ 1; 2Þ; respectively. By Lemma 3.1(2), if i; jAJs ðs ¼ 1; 2Þ with ioj; then Fðei #ej Þ ¼ aij fyð jÞ #fyðiÞ ; if iAJ1 ; jAJ2 ; then Fðei #ej Þ ¼ aij fyðiÞ #fyð jÞ ; where aij AC and jaij j ¼ 1; and for any finite rank operator F ABðH2 ; H1 Þ; FðF ÞABðK2 ; K1 Þ is a finite rank operator. Set Fs ¼ FjAlg Ns ; s ¼ 1; 2: By Lemma 3.6, Fs : Alg Ns -Alg Ms are additive bijective maps which preserve the closure of numerical range. Hence by Lemma 3.5(2), for each s ðs ¼ 1; 2Þ; there exists a conjugate unitary operator Us from Hs onto Ks satisfying Us ðN> s Þ ¼ Ms such that Fs ðTs Þ ¼ Us Ts Us for every Ts AAlg Ns : Let LCJ be an arbitrary finite subset and PL denote the projection from P H onto HL ¼ " L2 ¼ L-J2 ; then L ¼ L1 þ L2 iAL Hi : Set L1 ¼ L-J1 and T1 T12 and yðLÞ ¼ yðL1 Þ þ yðL2 Þ: For each T ¼ AAlg N; PL TPL ¼ 0 T2 PL1 T1 PL1 PL1 T12 PL2 : 0 PL2 T2 PL2 Note that FL ¼ FjPL Alg NPL can be regarded as an additive map from the upper triangular matrix algebra Tn onto Tn which preserves the numerical range, where % By using the result [11, Theorem 11] for Tn ; it is seen that the strictly upper n ¼ L: triangular parts of FL ðPL TPL Þ are uniquely determined by the map y : L-yðLÞ: T1 T12 Meanwhile Example 3.2 implies that, for each T ¼ ; if DðTÞ ¼ 0 T2 U1 T1 U1 U1 T12 U2 ; then W ðDðTÞÞ ¼ W ðTÞ: So, by the uniqueness of the 0 U2 T2 U2 expression of FL we must have ! U1 PL1 T1 PL1 U1 U1 PL1 T12 PL2 U2 FðPL TPL Þ ¼ : 0 U2 PL2 T2 PL2 U2 Now the weak continuity of F yields that ! U1 T1 U1 U1 T12 U2 FðTÞ ¼ ; 0 U2 T2 U2
8T ¼
T1
T12
0
T2
! AAlg N:
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If (ii) occurs, let H1 ¼ spanfej j jAJ1 gAN and K2 ¼ spanffk j kAK2 gAM: Then H1 and K2 determine, respectively, the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 : Obviously, Ns and Ms are maximal atomic nests on Hs and Ks (s ¼ 1; 2), respectively. Now, applying Lemma 3.5(1), Example 3.1, and a similar argument just as in case (i), we see that F has the form (4) stated in the theorem. The proof is completed. & By Examples 3.1, 3.2 and Theorem 3.9, the following corollary is immediate. Corollary 3.10. Let N and M be two maximal atomic nests on Hilbert spaces H and K; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective additive map. Then the following statements are equivalent: (1) F preserves the numerical range. (2) F preserves the closure of numerical range. (3) F has one of the four forms stated in Theorem 3.9. Compared with Theorem 2.3, we conjecture that the weak continuity assumption for F in Theorem 3.9 may be superfluous. However, we are not able to omit this assumption at present. Our next result says that, for a class of special nests, the conjecture is true. Let o denote the order type of the set N of natural numbers, and Wn o be the order type of N: Let fej gN j¼1 be a base of H; and let Nn ¼ j¼1 fej g; then N ¼ ff0g; Nn ðnX1Þ; Hg is a maximal atomic nest on H of the order type o þ 1; and N> has order type 1 þ o : All maximal atomic nests of order type o þ 1 or 1 þ o appear in this way. Theorem 3.11. Let N be a maximal atomic nest on H of the order type o þ 1 or 1 þ o : Assume that F : Alg N-Alg N is an additive surjective map. Then F preserves the closure of numerical range if and only if there exists a unitary operator UABðHÞ satisfying UðNÞ ¼ N such that FðTÞ ¼ UTU for every TAAlg N: Proof. Assume that N has order type 1 þ o : Let CðT tr Þ ¼ FðTÞtr for each TAAlg N; where T tr is the transpose of T with respect to the orthonormal base consisting of the unit vectors in atoms of N: Then C : Alg N> -Alg N> is additive and preserves the closure of numerical range. Therefore, it suffices to check the case that N has order type o þ 1: Suppose that N has order type o þ 1 and take an orthonormal base fej gjAJ of H consists of unit vectors from the atoms of N: Then J ¼ N and it is obvious that the map y : J-J stated in Lemma 3.1 is identity. By Lemma 3.5, there is a unitary operator UABðHÞ satisfying UðNÞ ¼ N such that FðF Þ ¼ UFU for all F AAlgF N: Let CðTÞ ¼ U FðTÞU for each TAAlg N: Then C : Alg N-Alg N is an additive surjective map preserving the closure of numerical range and CðF Þ ¼ F for every F AAlgF N: We claim that CðTÞ ¼ T for each TAAlg N; and therefore, the result holds.
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For any TAAlg N; let CðTÞ ¼ S: If SaT; then there are some n and m in N with mXn such that /Sem ; en Sa/Tem ; en S: Without loss of generality, we assume that n; m are the smallest numbers satisfying the above inequality. Write T ¼ PN PN Pn PN PN PN j¼1 i¼j tji ej #ei and S ¼ j¼1 i¼j sji ej #ei : Take F0 ¼ j¼1 i¼j tji ej #ei : Then F0 þ T ¼
N N X X j¼nþ1
F0 þ S ¼
n N X X j¼1
i¼j
tji ej #ei ;
i¼j
ðsji tji Þej #ei þ
N N X X j¼nþ1
sji ej #ei :
i¼j
Let a ¼ smn tmn ; then aa0: Choose mAC such that jmj4jjT þ F0 jj and jm þ aj ¼ jmj þ jaj: Let F ¼ men #en ; then wðT þ F0 þ F Þ ¼ jmj; but we always have wðS þ F0 þ F Þ4jmj; which contradicts to the fact that W ðT þ F0 þ F Þ ¼ W ðS þ F0 þ F Þ: Hence S ¼ T: & Theorem 3.11 implies that only the case (1) in Theorem 3.9 occurs when both N and M have the order type o þ 1 (or, 1 þ o ). In the rest of this section, we will further illustrate some examples to show that almost all possible combinations of four forms (1)–(4) stated in Theorem 3.9 may occur. (a) If N and M have the order type o þ 1 and 1 þ o ; respectively, then only the case (2) in Theorem 3.9 occurs. (b) If N and M have the order type o þ 1 þ n and 1 þ o ð¼ 1 þ o þ nÞ; respectively, then only the case (3) in Theorem 3.9 occurs. (c) If N and M have the order type o þ 1 þ n and o þ 1 ð¼ n þ o þ 1Þ; respectively, then only the case (4) in Theorem 3.9 occurs. (d) If both N and M have the order type 1 þ o þ o þ 1; then the cases (1) and (2) in Theorem 3.9 occur, but (3) and (4) do not occur. (e) If both N and M have the order type n þ o þ o þ 1; then the cases (1) and (3) in Theorem 3.9 occur, but (2) and (4) do not occur. (f) If both N and M have the order type o þ o þ 1; then the cases (1) and (4) in Theorem 3.9 occur, but (2) and (3) do not occur. (g) If N and M have the order type o þ o þ 1 and 1 þ o þ o ; respectively, then the cases (2) and (3) in Theorem 3.9 occur, but (1) and (4) do not occur. (h) If N and M have the order type n þ o þ o þ 1 and 1 þ o þ o þ n; respectively, then the cases (2) and (4) in Theorem 3.9 occur, but (1) and (3) do not occur. (i) If N and M have the order type ðo þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o Þ þ ð1 þ o þ o þ 1Þ and ð1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1Þ þ ðo þ 1 þ o Þ; respectively, then the cases (3) and (4) in Theorem 3.9 occur, but (1) and (2) do not occur.
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( j) If both N and M have the order type ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ; then the cases (1), (3) and (4) in Theorem 3.9 occur, but (2) does not occur. (k) If N and M have the order type ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ and ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ; respectively, then the cases (2)–(4) in Theorem 3.9 occur, but (1) does not occur. (l) If both N and M have the order type o þ 1 þ o ; then all cases (1)–(4) in Theorem 3.9 occur. We remark that if the cases (1)–(3) hold in Theorem 3.9, then (4) holds, too. Let J and K be the totally ordered index sets in the proof of Theorem 3.9. Then there exists an order isomorphism y : J-K; an anti-order isomorphism t : J-K; and there exist decompositions J ¼ J1 þ J2 and K ¼ K1 þ K2 ; and anti-order isomorphisms ds : Js -Ks ðs ¼ 1; 2Þ: Let Zs ¼ y 3 t1 3 ds ðs ¼ 1; 2Þ; it is clear that Zs : Js -Ks are order isomorphisms and K ¼ Z2 ðJ2 Þ þ Z1 ðJ1 Þ: This implies that form (4) occurs, too. Similarly forms (1), (2) and (4) together will imply (3) in Theorem 3.9. Hence fð1Þ; ð2Þ; ð3Þg3fð1Þ; ð2Þ; ð4Þg3fð1Þ; ð2Þ; ð3Þ; ð4Þg: So the above examples exhaust all possible combinations.
4. Additive maps preserving the closure of numerical range on atomic nest algebras The discussions in Sections 2 and 3 make it possible to characterize further the additive maps on atomic nest algebras which preserve the closure of the numerical ranges. Theorem 4.1. Let N and M be atomic nests on complex Hilbert spaces H and K with atoms fHj j jAJg and fKk j kAKg; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective additive map. Then F preserves the closure of numerical range if and only if one of the following statements is true: (1) There is a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M such that FðTÞ ¼ UTU for all TAAlg N: (2) There is a conjugate unitary operator U : H-K satisfying UðN> Þ ¼ M such that FðTÞ ¼ UT U for all TAAlg N: (3) There exist H1 AN; K1 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively; there exist conjugate unitary operators Us : Hs -Ks satisfying Us ðN> s Þ ¼ Ms ðs ¼ 1; 2Þ; such that FðTÞ ¼
U1 T1 U1
U1 T12 U2
0
U2 T2 U2
! ;
8T ¼
T1
T12
0
T2
! AAlg N:
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(4) There exist H1 AN; K2 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 ; respectively; there exist unitary operators Us : Hs -Ks satisfying Us ðNs Þ ¼ Ms ðs ¼ 1; 2Þ; such that ! ! U2 T2 U2 U2 T12 U1 T1 T12 FðTÞ ¼ ; 8T ¼ AAlg N: 0 U1 T1 U1 0 T2 Proof. By Examples 3.1 and 3.2, we need only to check the ‘‘only if’’ part. Assume that F preserves the closure of numerical range. It follows from Remark 2.1 and Theorem 2:30 that there exists a one–one and onto map y : J-K and the restriction of F to DN is a linear map from DN onto DM such that
F " Tj jAJ
¼ " Uyð jÞ Tjw Uyð jÞ ; jAJ
where Uyð jÞ is a unitary from Hj onto Kyð jÞ ; Tjw ¼ Tj or Tjtr : Hereafter, without loss of generality, we may regard Kyð jÞ as Hj : Hence Uyð jÞ ABðHj Þ and K ¼ H: It is certainly possible that for every jAJ we can take an orthonormal base feði;jÞ j iAJj g of Hj so that Jj is a totally ordered index set and there exists an anti-order isomorphism tj from Jj onto itself satisfying that t2j is the identity idj on Jj : Let the transpose Tjtr of Tj be taken with respect to such orthonormal base. Fix an anti-order isomorphism tj : Jj -Jj satisfying t2j ¼ idj and let Ej be the unitary operator on Hj determined by Ej eði;jÞ ¼ eðtj ðiÞ;jÞ : It is obvious that Ej2 ¼ Ij (the identity on Hj ). Put U ¼ "kAK Uk Dy1 ðkÞ ; where Dy1 ðkÞ ¼ Iy1 ðkÞ if Tyw1 ðkÞ ¼ Ty1 ðkÞ ; Dy1 ðkÞ ¼ Ey1 ðkÞ if Tyw1 ðkÞ ¼ Tytr1 ðkÞ : Then U is a unitary operator from H onto K satisfying UðMÞ ¼ M by noting that Kk ¼ Hy1 ðkÞ : Denote Tjf ¼ Ej Tjtr Ej and let CðÞ ¼ U FðÞU: It is clear that C is additive, surjective and preserves the closure of numerical range from z Alg N onto Alg M: Moreover, for T ¼ "jAJ Tj ADN ; CðTÞ ¼ "kAK Ty1 ðkÞ ; where z Ty1 ðkÞ ¼ Ty1 ðkÞ or Tyf1 ðkÞ : # ¼ S fði; jÞ j iAJ g be the totally ordered set with ði; jÞoðl; mÞ if and only Let J j jAJ # is defined similarly. Let N0 and M0 be the maximal if jom or iol when j ¼ m: K atomic nests generated by the bases # feði;jÞ j ði; jÞAJg
ð4:1Þ
# feðl;kÞ j ðl; kÞAKg;
ð4:2Þ
and
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respectively. Obviously, Alg N0 DAlg N; Alg M0 DAlg M and CðDN -Alg N0 Þ ¼ DM -Alg M0 ;
CðLN Þ ¼ LM ;
here LN ¼ fTADN j T is strictly lower triangular with respect to the maximal atomic base in ð4:1Þg and LM is one with respect to (4.2). By Lemma 3.1 and the weak continuity of C; it is also clear that C maps Alg N0 onto Alg M0 : Thus it follows from Theorem 3.9 that C takes one of the following forms when restricted to Alg N0 : (i) There is a unitary operator V ABðH; KÞ satisfying V ðN0 Þ ¼ M0 such that CðTÞ ¼ VTV for all TAAlg N0 : (ii) There is a conjugate unitary operator V : H-K satisfying V ððN0 Þ> Þ ¼ M0 such that CðTÞ ¼ VT V for all TAAlg N0 : (iii) There exist H1 AN0 ; K1 AM0 which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N0 ¼ N01 þ N02 ; M0 ¼ M01 þ M02 ; respectively; there exist conjugate unitary operators Vs : Hs -Ks satisfying Vs ððN0s Þ> Þ ¼ M0s ðs ¼ 1; 2Þ; such that ! ! V1 T1 V1 V1 T12 V2 T1 T12 CðTÞ ¼ ; 8T ¼ AAlg N0 ; 0 V2 T2 V2 0 T2 where Ts AAlg N0s ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: (iv) There exist H1 AN0 ; K2 AM0 which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N0 ¼ N01 þ N02 ; M0 ¼ M02 þ M01 ; respectively; there exist unitary operators Vs ABðHs ; Ks Þ satisfying Vs ðN0s Þ ¼ M0s ðs ¼ 1; 2Þ; such that ! ! V2 T2 V2 V2 T12 T1 T12 V1 CðTÞ ¼ ; 8T ¼ AAlg N0 ; 0 V1 T1 V1 0 T2 where Ts AAlg N0s ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: To see how to go back to the atomic nest algebras, say (iii) occurs. Obviously, one must request that H1 AN and K1 AM: Thus H1 and K1 determine nest decompositions N ¼ N1 þ N2 and M ¼ M1 þ M2 ; respectively. Recall that
C " Tj jAJ
¼ " Tyf1 ðkÞ ¼ " Ey1 ðkÞ Jy1 ðkÞ Ty1 ðkÞ Jy1 ðkÞ Ey1 ðkÞ ; kAK
kAK
ð4:3Þ
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where Jj is a conjugate unitary operator on Hj defined by 0 1 X X Jj @ xeði;jÞ A ¼ x% eði;jÞ : ði;jÞAJj
ði;jÞAJj
Note that Vs ðHj Þ ¼ Kyð jÞ for each jAJs ; s ¼ 1; 2: Let Vk ¼ Vs jH 1 y
ðkÞ
if kAKs : Then,
for any T ¼ T1 "T2 ¼ ð"jAJ1 Tj Þ"ð"jAJ2 Tj Þ ¼ "jAJ Tj ADN -Alg N0 ; we have ! V1 T1 V1 0 CðTÞ ¼ ð4:4Þ ¼ " Vk Ty1 ðkÞ Vk : 0 V2 T2 V2 kAK Comparing (4.3) with (4.4), one has Ey1 ðkÞ Jy1 ðkÞ Ty1 ðkÞ Jy1 ðkÞ Ey1 ðkÞ ¼ Vk Ty1 ðkÞ Vk and hence, Ey1 ðkÞ Jy1 ðkÞ Ty1 ðkÞ Jy1 ðkÞ Ey1 ðkÞ ¼ Vk Ty1 ðkÞ Vk for every kAK: Now it is easily checked that the formula in (iii) also holds when C is restricted to LN : Since FðÞ ¼ UCðÞU and UðMÞ ¼ M; we see that F takes form (3) stated in the theorem. The other cases are dealt similarly. The proof is complete. & For the linear case, we have the following result which in particular generalizes [2, Theorem 2.5]. Theorem 4.2. Let N and M be atomic nests on complex Hilbert spaces H and K; respectively. Suppose that F : Alg N-Alg M is a weakly continuous and surjective linear map. Then F preserves the closure of numerical range if and only if one of the following holds: (1) There is a unitary operator U : H-K satisfying UðNÞ ¼ M such that FðTÞ ¼ UTU for all TAAlg N: (2) There is a conjugate unitary operator U : H-K satisfying UðN> Þ ¼ M such that FðTÞ ¼ UT U for all TAAlg N:
5. Numerical radius isometries on atomic nest algebras Now we are in a position to give the main results of this paper promised in the introduction. Theorem 5.1. Let N and M be atomic nests on complex Hilbert spaces H and K; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective
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map. Then F is a numerical radius isometry if and only if there exists a complex unit c and an operator RAAlg M such that one of the following statements is true: (1) There is a unitary operator (or, conjugate unitary operator) U : H-K satisfying UðNÞ ¼ M such that FðTÞ ¼ cUTU þ R for all TAAlg N: (2) There is a conjugate unitary operator (or, unitary operator) U : H-K satisfying UðN> Þ ¼ M such that FðTÞ ¼ cUT U þ R for all TAAlg N: (3) There exist H1 AN; K1 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively; there exist conjugate unitary operators (or, unitary operators) Us : Hs -Ks satisfying Us ðN> s Þ ¼ Ms ðs ¼ 1; 2Þ such that
FðTÞ ¼ c
U1 T1 U1
U1 T12 U2
0
U2 T2 U2
! þ R;
8T ¼
T1
T12
0
T2
! AAlg N:
(4) There exist H1 AN; K2 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 ; respectively; there exist unitary operators (or, conjugate unitary operators) Us : Hs -Ks satisfying Us ðNs Þ ¼ Ms ðs ¼ 1; 2Þ such that U2 T2 U2 FðTÞ ¼ c 0
U2 T12 U1 U1 T1 U1
! þ R;
8T ¼
T1 0
T12 T2
! AAlg N:
Theorem 5.2. Let N and M be two atomic nests on H and K; respectively, and F : DN -DM be a surjective map. Then F is a numerical radius isometry if and only if there exist space decompositions H ¼ X1 "X2 "X3 "X4 and K ¼ Y1 "Y2 "Y3 "Y4 with Xs and Ys spanned by some atoms of N and M; respectively; there exist unitary operators Us : Xs -Ys ðs ¼ 1; 2Þ; conjugate unitary operators Vt : Xt -Yt ðt ¼ 3; 4Þ; a unitary operator CAZðDM Þ (the center of DM ), and an operator RADM such that FðTÞ ¼ CðU1 T1 U1 "U2 T2 U2 "V3 T3 V3 "V4 T4 V4 Þ þ R for every TADN ; where Ts ¼ TjXs ðs ¼ 1; y; 4Þ: We remark that, by the observations after the proof of Theorem 3.11, every possible combinations of (1)–(4) in Theorem 5.1 may occur, according to the choice of N and M: To prove above results, we need a lemma which was obtained by Li and Sˇemrl in [12] for the case of upper triangular block matrix algebras. The idea of our proof is also borrowed from there. Lemma 5.3. Let N be an atomic nest on complex Hilbert space H: Suppose S; TAAlg N are real linearly independent such that wðsS þ tTÞp1
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whenever s; tAR satisfying s2 þ t2 p1: Then the following conditions are equivalent: (1) There exists a complex unit m such that ðS; TÞ ¼ mðI; 7iIÞ: (2) For any AAAlg N; there exist s; tAR with s2 þ t2 ¼ 1 such that wðsS þ tT þ AÞ ¼ 1 þ wðAÞ: Proof. ð1Þ ) ð2Þ is obvious. We need only prove ð2Þ ) ð1Þ: Assume that (2) holds. Let fHj gjAJ be the set of all atoms of N: For jAJ and any xAHj with jjxjj ¼ 1; set Ay ¼ ðcos y þ i sin yÞx#x; where yA½0; 2pÞ: Then there exist sy ; ty AR such that s2y þ t2y ¼ 1 and wðsy S þ ty T þ Ay Þ ¼ 1 þ wðAy Þ ¼ 2: Thus, there exists a unit vector sequence fxn gN n¼1 CH such that lim j/ðsy S þ ty T þ Ay Þxn ; xn Sj ¼ 2:
n-N
Hence, we must have lim j/ðsy S þ ty TÞxn ; xn Sj ¼ 1
n-N
and
lim j/ðAy xn ; xn Sj ¼ 1:
n-N
According to the space decomposition H ¼ ½x"½x> ; write xn ¼ an x"un : Then 1 ¼ limn-N j/ðAy xn ; xn Sj ¼ limn-N jan j2 j/Ay x; xSj ¼ limn-N jan j2 : So there is a complex number a with jaj ¼ 1 such that limn-N xn ¼ ax; and consequently, 2 ¼ j/ðsy S þ ty TÞx; xS þ /Ay x; xSj and j/ðsy S þ ty TÞx; xSj ¼ j/Ay x; xSj ¼ 1: Therefore, /ðsy S þ ty TÞx; xS ¼ /Ay x; xS ¼ cos y þ i sin y:
ð5:1Þ
Suppose /Sx; xS ¼ h1 þ ih2 ; /Tx; xS ¼ k1 þ ik2 ; where h1 ; h2 ; k1 ; k2 AR: Let ! h1 k1 C¼ AM2 ðRÞ: h2 k2 By equality (5.1), for any yA½0; 2pÞ; there exists a unit vector uy ¼ ðsy ; ty Þtr AR2 such that Cuy ¼ ðcos y; sin yÞtr : Hence C maps the unit ball in R2 onto itself. Thus C is an isometry on R2 and is of the form ! ! cos t sin t cos t sin t C¼ or : sin t cos t sin t cos t
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It follows that /Tx; xS ¼ 7i/Sx; xS and j/Sx; xSj ¼ 1 for each unit vector xAHj ; jAJ: Fix xAHj ; write /Sx; xS ¼ m and suppose that /Tx; xS ¼ im (the case /Tx; xS ¼ im may be treated similarly). Without loss of generality, we may assume m ¼ 1: With respect to the space decomposition H ¼ ½x"½x> ; S and T have the following matrix representations:
S¼
1
S12
S21
S22
! and
T ¼i
1
T12
T21
T22
! :
Note that for any self-adjoint operator AABðHÞ satisfying wðAÞ ¼ 1; if there exists Ax0 ¼ x0 and according some x0 AH with jjx0 jj ¼ 1 such that /Ax0 ; x0 S ¼ 1; then 1 0 > to the space decomposition H ¼ ½x0 "½x0 ; A ¼ : Thus we observe that, 0 A22 for every BABðHÞ with wðBÞ ¼ 1; if there is a unit vector x0 AH such that /Bx0 ; x0 S ¼ 1; then BþB has the direct sum decomposition in term of the space 2 1 B12 decomposition H ¼ ½x0 "½x0 > ; and consequently, B ¼ : B12 B22 For any yA½0; 2pÞ; let Zy ¼ ðcos y þ i sin yÞ1 ðcos yS þ sin yTÞ ¼ ðcos y þ i sin yÞ
1
cos y þ i sin y cos yS21 þ i sin yT21
cos yS12 þ i sin yT12 cos yS22 þ i sin yT22
! :
By (5.1), we can apply the above observation to Zy and get ðcos y þ i sin yÞ1 ðcos yS12 þ i sin yT12 Þ ¼ ððcos y þ i sin yÞ1 ðcos yS21 þ i sin yT21 ÞÞ ; ¼ S21 : so T12 ¼ S12 ¼ T21 For arbitrary unit vector yAH orthogonal to x satisfying y#yAAlg N; there is some kAJ so that yAHk : Without loss of generality, assume that jpk: For the sake of convenience, we write a ¼ /Sy; xS; l ¼ /Sy; yS; then /Sx; yS ¼ a% ; /Ty; yS ¼ dðiÞl; where dðiÞ ¼ 7i: For every yA½0; 2pÞ; we have
Zy;y ¼ P½x;y ðcos yS þ sin yTÞj½x;y ¼e
iy
1
a
a% eiy ðcos y þ dðiÞsin yÞl
! :
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Suppose aa0: If dðiÞ ¼ i; then there exists a yA½0; 2pÞ such that l% aei2y al% a% ei2y ; thus l% eiy Zy;y ¼
l% ei2y l% a% ei2y
l% aei2y
!
1
has numerical radius one and 12ðl% eiy Zy;y þ ðl% eiy Zy;y Þ Þ has an eigenvalue larger than one, which is a contradiction. Hence we have /Ty; yS ¼ il and Zy;y ¼ eiy
1
a
a%
l
! :
Let x be a complex unit such that ax ¼ jaj and let Bx ¼ xx#x þ xy#y þ 2x#y: Then W ðBx Þ is a circle disk centered at x with radius 1 and hence, wðBx Þ ¼ 2: By condition (2), there exists a yA½0; 2pÞ such that wðcos yS þ sin yT þ Bx Þ ¼ 3: A similar argument previous to (5.1) shows that there is a unit vector vAspanfx; yg such that 3 ¼ j/ðcos yS þ sin yTÞv; vS þ /Bx v; vSj ¼ j/ðcos yS þ sin yTÞv; vSj þ j/Bx v; vSj: Note that j/Bx v; vSj ¼ 2 if and only if v ¼ Zðx þ xyÞ for some ZAC with jZj ¼ Hence * ! + 1 þ l 1 a 1 ; 1 ¼ j/ðcos yS þ sinyTÞv; vSj ¼ v; v ¼ 2 2 a% l
pffiffi 2 2 :
and consequently, l ¼ 1: Therefore /Sy; yS ¼ /Sx; xS ¼ 1 and /Ty; yS ¼ i: Suppose a ¼ 0: Let xAf1; ig and Bx ¼ xðx#x þ y#yÞ þ 2x#y; then wðBx Þ ¼ 2: Now, similar to the previous argument, there exist s; tAR with s2 þ t2 ¼ 1 and a unit vector vAspanfx; yg such that 3 ¼ wðsS þ tT þ Bx Þ ¼ j/ðsS þ tT þ Bx Þv; vSj ¼ j/ðsS þ tTÞv; vS þ /Bx v; vSj ¼ j/ðsS þ tTÞv; vSj þ j/Bx v; vSj;
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hence /ðsS þ tTÞv; vS and /Bx v; vS have the same direction. If x ¼ 1; then /Bx v; vS ¼ 2 and 1 ¼ /ðsS þ tTÞv; vS ¼
ðs þ itÞ þ ðs7itÞl : 2
This entails that both complex units s þ it and ðs7itÞl are equal to 1, and consequently, s ¼ 1; t ¼ 0 and l ¼ 1: So /Sy; yS ¼ /Sx; xS ¼ 1: If x ¼ i; then /Bx v; vS ¼ 2i and i ¼ /ðsS þ tTÞv; vS ¼
ðs þ itÞ þ ðs7itÞ : 2
Thus s ¼ 0; t ¼ 1 and /Ty; yS ¼ i/Sy; yS ¼ i: Up to now we have proved, for arbitrary orthogonal unit vectors xAHj and yAHk with jpk; that /Sx; xS ¼ /Sy; yS with j/Sx; xSj ¼ 1 and /Ty; yS ¼ /Tx; xS ¼ i/Sx; xS: For every jAJ; let Pj denote the projection from H onto Hj : The convexity of numerical range, together with the fact that j/Sx; xSj ¼ 1 for all unit vectors xAHj ; implies that there exists a unit complex number mj such that Pj SPj ¼ mj Pj : Since, for any unit vectors xAHj and yAHk ; /Sx; xS ¼ /Sy; yS; there must be a constant m with jmj ¼ 1 such that mj ¼ m for all jAJ: We claim that S ¼ mI: Otherwise, there would exist some unit vectors xAHj and yAHk with jok so that m g g ¼ /Sy; xSa0: Thus S has a principal submatrix whose numerical radius 0 m is greater than jmj ¼ 1 ¼ wðSÞ; a contradiction. Thereby, for any unit vector xAHj ð8jAJÞ; we have /Tx; xS ¼ i/Sx; xS ¼ im and a similar argument implies T ¼ imI: & Proof of Theorem 5.1. We divide the proof into several steps. Step 1: For every TAAlg N; let CðTÞ ¼ FðTÞ Fð0Þ: Then C : Alg N-Alg M is a weakly continuous and surjective real linear map preserving numerical radius. It is obvious that Cð0Þ ¼ 0; C is weakly continuous, surjective and wðCðTÞÞ ¼ wðFðTÞ Fð0ÞÞ ¼ wðTÞ for every TAAlg N: Since the numerical radius is a norm, C is real linear from the Mazur–Ulam theorem [14] which states that every surjective isometry f satisfying fð0Þ ¼ 0 from a normed space onto another one is real linear. Step 2: There exists a complex unit c such that CðIÞ ¼ cI and CðiIÞ ¼ 7icI: For every AAAlg N; there exist some s; tAR with s2 þ t2 ¼ 1 such that wðsI þ itI þ AÞ ¼ 1 þ wðAÞ: Since C is real linear and preserves the numerical radius, we have wðsCðIÞ þ tCðiIÞ þ CðAÞÞ ¼ 1 þ wðCðAÞÞ for every AAAlg N: The surjectivity of C implies that ðCðIÞ; CðiIÞÞ satisfies condition (2) in Lemma 5.3, so there exists a cAC with jcj ¼ 1 such that CðIÞ ¼ cI and CðiIÞ ¼ 7icI: Without loss of generality, we assume that CðIÞ ¼ I and CðiIÞ ¼ 7iI: Hence, either CðlIÞ ¼ lI for all lAC; or CðlIÞ ¼ l% I for all lAC: Step 3: Assume that CðlIÞ ¼ lI for all lAC; then C preserves the closure of numerical range.
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For any TAAlg N; W ðTÞ is bounded and convex, so W ðTÞ is a compact convex subset in C: Assume that there exists a mAC such that mAW ðCðTÞÞ\W ðTÞ; then, there exists a circle with sufficiently large radius centered at a certain lAC such that W ðTÞ lies inside the circle, but m lies outside the circle. Hence, for any zAW ðTÞ; we have jz ljojm lj: Consequently, by the real linearity of C and the fact that CðlIÞ ¼ lI; one gets wðT lIÞojm ljpwðCðTÞ lIÞ ¼ wðCðT lIÞÞ ¼ wðT lIÞ; which is a contradiction. So W ðCðTÞÞDW ðTÞ and hence W ðCðTÞÞDW ðTÞ: Using the same argument to C1 ; we see that W ðTÞDW ðCðTÞÞ: It follows that, for all TAAlg N; W ðCðTÞÞ ¼ W ðTÞ; that is, C preserves the closure of numerical range. Step 4: Assume that for any lAC; CðlIÞ ¼ l% I: Let jðTÞ ¼ CðTÞ for every TAAlg N: Then j : Alg N-Alg M> is a weakly continuous and surjective additive map which preserves the closure of numerical range. It is clear that j is a weakly continuous and surjective additive map which preserves the numerical radius. Since jðlIÞ ¼ lI for each lAC; by the proof of Step 3, we have j : Alg N-Alg M> preserves the closure of numerical range. Now, applying Theorem 4.1 to C in Step 3 and to j in Step 4, it is easily seen that F has one of the eight forms stated in Theorem 5.1 and the proof is finished. To make the point clearer, we assume, for example, that j has form (1) in Theorem 4.1. That is, there exists a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M> such that jðTÞ ¼ UTU for every TAAlg N: Then UðN> Þ ¼ M and CðTÞ ¼ UT U for every TAAlg N; which is one of two forms stated in (2) of Theorem 5.1. & We remark that, by the observations after the proof of Theorem 3.11, all possible combinations of forms (1)–(4) in Theorem 5.1 may occur. Particularly, in many cases, the nest N may not be unitarily equivalent to the nest M> ; or N and M may not have the decompositions described in (3) or (4) of Theorem 5.1. Thus the classification of numerical radius isometries from Alg N onto Alg M may sometimes have a neater expression. The following corollary is an immediate consequence of Theorems 3.11 and 5.1, which omits the assumption of weak continuity. Corollary 5.4. Let N be an atomic nest on Hilbert space H with the order type o þ 1 or 1 þ o and let F : Alg N-Alg N be a surjective map. Assume that every atom of N is finite dimensional. Then F is a numerical radius isometry if and only if there exists a unitary operator or conjugate unitary operator U on H satisfying UðNÞ ¼ N; an operator RAAlg N and a complex unit c such that FðTÞ ¼ cUTU þ R holds for every TAAlg N: In other words, Corollary 5.4 says that, if N be an atomic nest on Hilbert space H with the order type o þ 1 or 1 þ o and if its every atom is of finite dimension, then
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the set of all numerical radius isometries on Alg N is a group generated by the following simple maps: (1) T/UTU ; where U : H-H is a unitary or conjugate unitary operator satisfying UðNÞ ¼ N; (2) T/cT; where cAC and jcj ¼ 1; (3) T/T þ R; where RAAlg N: Next, we turn to the proof of Theorem 5.2. Proof of Theorem 5.2. Let fHj j jAJg and fKk j kAKg be the sets of all atoms of N and M; respectively. Then DN ¼ "jAJ BðHj Þ and DM ¼ "kAK BðKk Þ: Notice that CAZðDM Þ if and only if there exists a set fmk gkAK of complex numbers such that C ¼ "kAK mk Ik : Thus, for every GADM ; we have wðCGÞ ¼ wðGÞ if C is also unitary. So, the sufficiency of the condition is obvious and we need only verify the necessity. Assume that F is a numerical radius isometry. For each TADN ; let CðTÞ ¼ FðTÞ Fð0Þ; then C is a real linear map preserving the numerical radius. Claim. Suppose S; TADM are real linearly independent such that wðsS þ tTÞp1 whenever s; tAR satisfying s2 þ t2 p1: Then the following conditions are equivalent: (i) There exist complex units fmk j kAKg such that ðS; TÞ ¼ " mk ðIk ; 7iIk Þ; where Ik denotes the identity on Kk : kAK
(ii) For any AADM ; there exist s; tAR with s2 þ t2 ¼ 1 such that wðsS þ tT þ AÞ ¼ 1 þ wðAÞ: To prove the claim, write S ¼ "kAK Sk and T ¼ "kAK Tk ; where Tk ; Sk ABðKk Þ ðkAKÞ: Suppose that (i) holds. Let A ¼ "kAK Ak ADM be arbitrary. It is clear that, for any e40; there exists a kAK such that wðAÞowðAk Þ þ e: Using Lemma 5.3 for Ak ; we get that there exist s; tAR with s2 þ t2 ¼ 1 such that 1 þ wðAÞo 1 þ wðAk Þ þ e ¼ wðsSk þ tTk þ Ak Þ þ e p wðsS þ tT þ AÞ þ ep1 þ wðAÞ þ e: Let e-0; we see that condition (ii) holds. Conversely, suppose that (ii) holds. For any k0 AK; take Aðk0 Þ ¼ "kAK Ak ADM ; where Ak ¼ 0 if kak0 : Then 1 þ wðAk0 Þ ¼ 1 þ wðAðk0 Þ Þ ¼ wðsS þ tT þ Aðk0 Þ Þ ¼ wðsSk0 þ tTk0 þ Aðk0 Þ Þp1 þ wðAk0 Þ; so wðsSk0 þ tTk0 þ Aðk0 Þ Þ ¼ 1 þ wðAk0 Þ: Now by Lemma 5.3, there exists a complex unit mk0 such that ðSk0 ; Tk0 Þ ¼ mk0 ðIk0 ; 7iIk0 Þ: Therefore we have ðiiÞ ) ðiÞ: Hence Claim holds.
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Since for any BADN ; there exist s; tAR with s2 þ t2 ¼ 1 such that wðsI þ itI þ BÞ ¼ 1 þ wðBÞ; we have wðsCðIÞ þ tCðiIÞ þ CðBÞÞ ¼ 1 þ wðCðBÞÞ by the numerical radius preservation and real linearity of C: Now Claim, together with the surjectivity of C; implies that there exist complex units mk ð8kAKÞ such that ðCðIÞ; CðiIÞÞ ¼ "kAK mk ðIk ; 7iIk Þ: Let jðTÞ ¼ CðIÞ1 CðTÞ for every TADN : Since CðTÞ has the form CðTÞ ¼ "kAk Ek ; we have jðTÞ ¼ "kAk mk Ek and wðjðTÞÞ ¼ supkAk fwðmk Ek Þg ¼ wðCðTÞÞ ¼ wðTÞ: Hence j : DN -DM is a unital real linear map preserving the numerical radius. Let H 1 ¼ fkAK j jðiIÞk ¼ iIk g and H 2 ¼ fkAK j jðiIÞk ¼ iIk g: Then H 1 ,H 2 ¼ K and jðiIÞ ¼ ð"kAH 1 iIk Þ"ð"kAH 2 ðiIk ÞÞ: Define p : DM -DM by pð"kAK Gk Þ ¼ ð"kAH 1 Gk Þ"ð"kAH 2 Gk Þ for every "kAK Gk ADM : It is obvious that p is a real linear bijection preserving the numerical radius. Let c ¼ p 3 j; then c : DN -DM is a real linear surjective map which preserves the numerical radius and cðlIÞ ¼ lI for each lAC: A similar argument just as in Step 3 of the proof of Theorem 5.1 implies that c preserves the closure of numerical range. Now it follows from Theorem 2.3 and its proof that there exists a 1-1 and onto map y : J-K; there exist subsets J01 and J02 of J with J ¼ J01 ,J02 ; and there exist unitary operators Uj ABðHj ; Kyð jÞ Þ if jAJ01 ; conjugate unitary operators Vj : Hj -Kyð jÞ if jAJ02 ; such that for every T ¼ "jAJ Tj ADN ; ! ! cðTÞ ¼
" Uy1 ðkÞ Ty1 ðkÞ Uy1 ðkÞ "
kAK01
" Vy1 ðkÞ Ty1 ðkÞ Vy1 ðkÞ ;
kAK02
where K0s ¼ yðJ0s Þ; s ¼ 1; 2: Let Ks ¼ K01 -H s and K2þt ¼ K02 -H t ; s; t ¼ 1; 2: Then for every T ¼ "jAJ Tj ADN ; we have jðTÞ ¼ " Uy1 ðkÞ Ty1 ðkÞ Uy1 ðkÞ " " Uy1 ðkÞ Ty1 ðkÞ Uy1 ðkÞ kAK1 kAK2 " " Vy1 ðkÞ Ty1 ðkÞ Vy1 ðkÞ " " Vy1 ðkÞ Ty1 ðkÞ Vy1 ðkÞ : kAK3
kAK4
Put C ¼ CðIÞ and R ¼ Fð0Þ; then we see that F has the desired form.
&
References [1] Z.F. Bai, J.C. Hou, Numerical radius distance preserving maps on BðHÞ; Proc. Amer. Math. Soc., to appear. [2] J.L. Cui, J.C. Hou, Linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest, Integer Equation and Operator Theory, to appear. [3] K.R. Davidson, Nest Algebra, in: Ritman Research Notes in Mathematics, Vol. 191, Longman, London, New York, 1988. [4] G. Frobenius, Uber die darstellung der endlichen gruppen durch lineare subsitutionen, I., Sitzungberichte Koniglich Preussischen Akad. Wissenschaften Berlin, 1897, pp. 994–1015.
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[5] P.R. Halmos, A Hilbert Space Problem Book, 2nd Edition, Springer, New York, 1982. [6] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991. [7] L.K. Hua, Geometry of matrices I. Generalizations of von staudt’s theorem, Trans. Amer. Math. Soc. 57 (1945) 441–481. [8] L.K. Hua, Geometry of matrices II. Study of involutions in the geometry of symmetric matrices, Trans. Amer. Math. Soc. 61 (1947) 193–228. [9] L.K. Hua, Geometry of matrices III. Fundamental theorems in the geometry of symmetric matrices, Trans. Amer. Math. Soc. 61 (1947) 229–255. [10] L.K. Hua, Geometry of symmetric matrices over any field with characteristic other than two, Ann. Math. 50 (1949) 8–31. [11] G. Lesnjak, Additive preserves of numerical range, Linear Algebra Appl. 345 (2002) 235–253. [12] C.K. Li, P. Semrl, Numerical radius isometries, Linear and Multilinear Algebras 50 (2002) 307–314. [13] C.K. Li, N.K. Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162–164 (1992) 217–235. [14] S. Mazur, S. Ulam, Sur les transformations isom’etriques d’espaces vectoriels norm’es, C. R. Acad. Sci. Paris 194 (1932) 946–948. [15] Z.X. Wan, Geometry of Matrices, World Scientific, Singapore, 1996.
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Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras$ Jianlian Cuia,1 and Jinchuan Houb,,2 b
a School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China Department of Mathematics, Shanxi Teachers University, Linfen, 041004, People’s Republic of China
Received 30 September 2002; revised 22 December 2002; accepted 15 January 2003 Communicated by G. Pisier
Abstract A nonlinear map f between operator algebras is said to be a numerical radius isometry if wðfðT SÞÞ ¼ wðT SÞ for all T; S in its domain algebra, where wðTÞ stands for the numerical radius of T: Let N and M be two atomic nests on complex Hilbert spaces H and K; respectively. Denote Alg N the nest algebra associated with N and DN ¼ Alg N-ðAlg NÞ the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from Alg N onto Alg M and a thorough classification of numerical radius isometries from DN onto DM : r 2003 Elsevier Inc. All rights reserved. MSC: Primary 47H20; 47L35; 47A12; Secondary 47B48 Keywords: Numerical radius; Non-linear isometries; Nest algebras
1. Introduction Let H and K be complex Hilbert spaces with inner product / ; S; and BðH; KÞ (BðHÞ if H ¼ K) be Banach space of all bounded linear operators from H into K: Assume that A and B are subalgebras (or subspaces) in BðHÞ and BðKÞ; $
This work was partly supported by NNSFC and PNSFS. Corresponding author. E-mail address: [email protected] (J. Hou). 1 Department of Applied Mathematics, Taiyuan University of Technology, Taiyuan 030024, People’s Republic of China; Department of Mathematics, Shanxi Teachers University, Linfen, 041004, People’s Republic of China. 2 Department of Mathematics, Shanxi University, Taiyuan 030000, People’s Republic of China.
0022-1236/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-1236(03)00076-4
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respectively, and F : A-B is a map (maybe non-linear). A general and interesting problem that should be considered is as follows: Problem 0.1. Find as few as possible properties that may be possessed by A and B or by elements in them and that are enough to determine the structure of the map F if F takes these properties as invariants, i.e., if F preserves these properties. When F is linear, the above problem is so-called the linear preserver problem, of which the study can be traced back to the work of Frobenius [4] and has become one of the most active and fertile research fields in the matrix theory and operator theory during the past decades (see the survey paper [13]). Many results concerning linear preserver problems reveal the relations between linear structure and algebraic structure of operator algebras. Problem 0.1 is also associated with the geometry of matrices whose study was pioneered by Hua in the 1940s (see [7–10]). Let F be a field and Mnn ðFÞ be the space of all n n matrices over F: Consider the group of motions on Mnn ðFÞ consisting of the following maps: T/PTQ þ R;
8TAMnn ðFÞ
or
T/PT tr Q þ R;
8TAMnn ðFÞ;
where P; Q and R are n n matrices with P and Q being nonsingular, T tr denotes the transpose matrix of T: The fundamental problem of geometry of matrices is to characterize the group of motions (to within automorphisms of the underlying field) by as few geometry invariants as possible [15]. Hua proved that ‘‘adjacency’’ (T and S are adjacent if rankðT SÞ ¼ 1) is such an invariant for Mnn ðFÞ over some field F; especially the real field R and the complex field C: Thus our Problem 0.1 asks indeed the possibility of developing an analog of ‘‘geometry of matrices’’ for operators. The numerical range and numerical radius of TABðHÞ are respectively defined as W ðTÞ ¼ f/Tx; xS j xAH and jjxjj ¼ 1g; wðTÞ ¼ supfjlj j lAW ðTÞg: These concepts and their generalizations have been studied extensively because of their connections and applications to many different areas. Bai and Hou [1] found that the geometry invariant ‘‘numerical radius distance’’ alone is sufficient to characterize the nonlinear maps from BðHÞ onto BðKÞ: Precisely, it was shown that F : BðHÞ-BðKÞ is a surjective numerical radius isometry, i.e., wðFðTÞ FðSÞÞ ¼ wðT SÞ holds for all T; SABðHÞ if and only if there exist complex unit m (i.e., mAC and jmj ¼ 1) and operator RABðKÞ such that F takes one of the following (four) forms: (1) There exists a unitary or conjugate unitary operator U : H-K such that FðAÞ ¼ mUAU þ R for every AABðHÞ:
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(2) There exists a unitary or conjugate unitary operator U : H-K such that FðAÞ ¼ mUA U þ R for every AABðHÞ: Therefore the group of all nonlinear surjective numerical radius isometries on BðHÞ is generated by the following five kinds of very simple maps (motions): (i) (ii) (iii) (iv) (v)
T/UTU ; where U : H-H is a unitary operator; T/UTU ; where U : H-H is a conjugate unitary operator; T/T ; T/cT; where cAC and jcj ¼ 1; T/T þ R; where RABðHÞ:
Note that, it follows from the above result that every (nonlinear) numerical radius isometry from BðHÞ onto BðKÞ is also an operator norm isometry, but the converse proposition fails. In this paper, we consider the same question of characterizing the nonlinear surjective numerical radius isometries for more general cases, i.e., for the nest algebra case and the diagonal algebra case. Let N and M be atomic nests on H and K; respectively. Denote the associated nest algebras by Alg N and Alg M; and the associated diagonal algebras by DN and DM ; respectively. In [2], we characterized the linear maps preserving the numerical radius from DN onto DM and, when N and M are maximal atomic, characterized the weakly continuous linear maps preserving the numerical radius from Alg N onto Alg M: In this paper, we find further that the invariant ‘‘numerical radius distance’’ is also enough to determine the structure of nonlinear maps from DN onto DM and the structure of weakly continuous nonlinear maps from Alg N onto Alg M; and then give their complete classifications. Not like the case of BðHÞ; the group of surjective numerical radius isometries on a nest algebra or a diagonal algebra may has other kinds of generators which are no longer so nicely to keep the algebraic and geometric structures. By Mazur–Ulam theorem [14], to achieve our goal, one of the keys is to reduce the question to that of characterizing the additive maps which preserve the closure of numerical range. For finite dimensional cases, that is, for the cases of upper triangular block matrix algebra T and diagonal block matrix algebra D; the additive maps preserving the numerical range were characterized by Lesnjak [11] and, based on Lesnjak’s work, the classifications of numerical radius isometries on T and D were given by Li and Sˇemrl [12]. The results and the basic ideas in [11,12] are enlightening and useful to our study here. The paper is organized as follows. In Section 2, we give some basic properties of the additive maps which preserve the closure of numerical range from an atomic nest algebra onto another one, and get a characterization of such maps when restricted to the diagonal algebras. Section 3 is the crucial parts of this paper, in which we get a complete description of weakly continuous additive maps preserving the closure of numerical range from a maximal atomic nest algebra onto another one. In Section 4, we generalize the results in Section 3 to the atomic nest algebra case. Based on the discussions in previous sections, Section 5 is devoted to characterizing the numerical
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radius isometries. The main results are Theorems 5.1 and 5.2, which give respectively a complete characterization of the weakly continuous numerical radius isometries between atomic nest algebras and the numerical radius isometries between diagonal algebras. Comparing Theorem 5.1 with Theorem 5.2, we need the additional assumption of weak continuity for maps (even for linear maps, see Theorem 4.2 and [2]) on nest algebras. We guess that the assumption of weak continuity in Theorem 5.1 is not necessary, and we answer affirmatively this conjecture in some situations (See Corollary 5.4). But in this paper we are not able to omit this assumption for general cases. Now we fix some notations. Recall that a nest on H is a chain N of closed (under norm topology) subspaces of H containing f0g and H; which W is closed under the formationV of arbitrary closed linear span (denoted by ) and intersection (denoted by ). Alg N denotes the associated nest algebra, which is the set of all operators T in BðHÞ such that TNDN for every element NAN: We denote AlgF N ¼ Alg N-FðHÞ; the set of all finite rank operators in Alg N: Relation ‘‘ACB’’ means that A is a proper subset of B: The self-adjoint operator algebra DN ¼ Alg N-ðAlg WNÞ is called the diagonalValgebra associated with N: For NAN; define N ¼ fMAN j MCNg; Nþ ¼ fMAN j NCMg and define 0 ¼ 0; Hþ ¼ H: As usual, N > is the orthogonal complement of N: If N~N ¼ N-ðN Þ> a0; we say N~N is an atom of N: A nest N on H is said to be atomic if H is spanned by its atoms, and to be maximal atomic if N is atomic and all its atoms are one dimensional (Ref. [3]). Let x; f AH; the rank-1 operator defined by y//y; f Sx will be denoted by x#f : Two nests N and M on H and K; respectively, is unitarily equivalent if there is a unitary operator (or equivalently, a conjugate unitary operator) U : H-K such that UðNÞ ¼ fUðNÞ j NANg ¼ M: Recall that U is a conjugate unitary operator if U is conjugate linear (i.e., U is additive and UðlxÞ ¼ l% Ux for any lAC and xAH) and both UU and U U are the identity operators. Note that, for a conjugate linear operator A on H; one has /Ax; yS ¼ /A y; xS for all x; yAH: A projection PABðHÞ is a self-adjoint and idempotent operator (i.e., P ¼ P ¼ P2 ). We denote PL the projection whose range is the subspace L: As usual, C; R and N stand for the complex number field, real number field and natural number set, respectively. At the end of introduction, we recall some basic results on numerical range and numerical radius used often in our study. One may see [5] and Chapter 1 of [6] for more information. Proposition 1.1 (See Halmos [5]). Let AABðHÞ: (1) W ðAÞ ¼ W ðAtr Þ; where Atr stands for the transpose of A relative to an arbitrary orthonormal base of H: (2) W ðAÞ ¼ W ðUAU Þ for any unitary UABðHÞ: (3) W ðlAÞ ¼ lW ðAÞ for any lAC: (4) W ðlI þ AÞ ¼ l þ W ðAÞ for any lAC:
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Proposition 1.2 (See Halmos [5]). The numericalrange ofAABðHÞ is always convex. l1 b ; then W ðAÞ is an elliptical In particular, if AAM2 ðCÞ is unitarily similar to 0 l2 disk with l1 and l2 as foci, and the length of minor axis equal to jbj; where M2 ðCÞ denotes the 2 2 complex matrix algebra. Proposition 1.3 (See Halmos [5]). Let AABðHÞ: Then W ðAÞ ¼ flg if and only if A ¼ lI: Proposition 1.4 (See Halmos [5], Horn and Johnson [6]). Suppose that NCH is a closed subspace and AABðHÞ: Then W ðPN AjN ÞDW ðAÞ and wðPN AjN ÞpwðAÞ: Proposition 1.5 (See Horn and Johnson [6, Section 1.2.9]). Suppose that AAMn ðCÞ such that A þ A (resp., AA i ) have ln and l1 as the largest and smallest eigenvalues, respectively. Then n z z o % ½l1 ; ln ¼ fz þ z% j zAW ðAÞg resp:; ½l1 ; ln ¼ zAW ðAÞ : i Proposition 1.6 (See Halmos [5]). If AABðHÞ is unitarily similar to A1 "A2 ; then W ðAÞ ¼ convfW ðA1 Þ,W ðA2 Þg; where convðOÞ denotes the convex hull of the set O:
2. Additive maps preserving the closure of numerical range on diagonal algebras Let N be an atomic nest on Hilbert space H: Assume that fHi j iAJg is the set of all atoms of N; then H ¼ "iAJ Hi : There is a natural order ‘‘p’’ so that J becomes a totally ordered set. For any i; jAJ; ipj if and only if xi #yj AAlg N whenever xi AHi and yj AHj : ioj means ipj and iaj: It is clear in this case that DN ¼ "iAJ BðHi Þ: We say that two projections P; QAAlg N are orthogonal if PQ ¼ 0: Lemma 2.1. Let N and M be two atomic nests on Hilbert spaces H and K; respectively. Assume F : Alg N-Alg M (or F : DN -DM ) is an additive surjective map. If F preserves the closure of numerical range, then F preserves the orthogonal projections of rank one in both directions. Furthermore, FðlPÞ ¼ lFðPÞ for every rank-1 projection P and lAC: Proof. Since F preserves the closure of numerical range, it is easy to see that F is injective and preserves the numerical radius. Note that the numerical radius is a
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norm which is equivalent to the usual operator norm, so F is continuous and hence is real linear. For any finite rank operator F ; we have W ðF Þ ¼ W ðF Þ: Let PAAlg N be an arbitrary rank-1 projection. Then ½0; 1 ¼ W ðPÞ ¼ W ðPÞ ¼ W ðFðPÞÞ; and therefore, FðPÞX0: Set FðPÞ ¼ B þ C; where B and C are positive operators in Alg M; then P ¼ F1 ðBÞ þ F1 ðCÞ; here F1 ðBÞ and F1 ðCÞ are positive since F1 also preserves the closure of numerical range. Lemma 2.1 in [2] states that a positive operator A in a nest algebra is a scalar multiple of a projection of rank one if and only if A ¼ B þ C with B and C being positive in the nest algebra implies that both B and C are scalar multiples of A: Thus, there is a real number rA½0; 1 such that F1 ðBÞ ¼ rP and F1 ðCÞ ¼ ð1 rÞP: So B ¼ rFðPÞ and C ¼ ð1 rÞFðPÞ by the real linearity of F: Consequently, FðPÞ is a positive multiple of a projection of rank one. Since W ðFðPÞÞ ¼ W ðFðPÞÞ ¼ ½0; 1; FðPÞ is a rank-1 projection. Now suppose that rank-1 projections P; QAAlg N are orthogonal. Let FðPÞ ¼ x#x and FðQÞ ¼ y#y: If x and y belong to the different atoms of M; then it is clear that FðPÞ and FðQÞ are orthogonal. Otherwise, there exists MAM with dimðM~M ÞX2 such that x; yAM~M : Since fj/z; xSj2 j/z; ySj2 j jjzjj ¼ 1g ¼ W ðFðP QÞÞ ¼ W ðP QÞ ¼ ½1; 1; and since j/z; xSj2 j/z; ySj2 attains the largest value 1 only when z and x are linearly dependent and z>y; we must have /x; yS ¼ 0: So FðPÞ and FðQÞ are orthogonal. Because F1 has the same properties as F has, F preserves the orthogonal projections of rank one in both directions. Let u#uAAlg N be a projection, Fðu#uÞ ¼ x#x and let Fðiu#uÞ ¼ B; where i denotes the imaginary unit. For any unit vector y in some atom of M such that y>x; there is a unit vector v in some atom of N such that Fðv#vÞ ¼ y#y: Note that v>u: It is clear that, for a sufficiently large positive number M; we have wðiu#u þ Mv#vÞ ¼ M: Thus wðB þ My#yÞ ¼ M as F preserves the numerical radius. It follows that j/By; yS þ MjpM; and consequently, /By; yS ¼ 0 by virtue of iBX0: Therefore, B must be a scalar multiple of x#x: Now it is obvious that Fðiu#uÞ ¼ ix#x and F is complex linear when restricted to the one dimensional space spanned by a rank-1 projection. The proof for the case F : DN -DM is just the same. & Lemma 2.2. Let N and M be two atomic nests on Hilbert spaces H and K; respectively, and let fHj j jAJg and fKk j kAKg be the set of all atoms of N and M; respectively. Assume F : Alg N-Alg M (or F : DN -DM ) is an additive surjective map. If F preserves the closure of numerical range, then there exists a 1-1 and onto map y : J-K such that for every jAJ; FðBðHj ÞÞ ¼ BðKyð jÞ Þ; here we identify BðHj Þ as a subalgebra of DN in an obvious way.
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Proof. Let jAJ: By Lemma 2.1, we need only to consider the case that the dimension of Hj is greater than one. Assume that u; vAHj are orthogonal unit vectors. By Lemma 2.1, there exist orthogonal unit vectors x and y such that Fðu#uÞ ¼ x#x and Fðv#vÞ ¼ y#y: For any lAC; since u þ lvAHj ; we have Fððu þ lvÞ#ðu þ lvÞÞ ¼ yl #yl ; where yl AKk for some kAK and jjyl jj2 ¼ jlj2 þ 1: If dim K ¼ 2; then yl is a linear combination of the vectors x and y; and hence, both x and y are in Kk : Assume that dim K42: For any nonzero vector z lying in some atom of M orthogonal to both x and y; by Lemma 2.1, there is a nonzero vector w belonging to some atom of N orthogonal to both u and v such that Fðw#wÞ ¼ z#z: It follows from the orthogonality of w and u þ lv that z>yl : This entails again that yl is a linear combination of x and y: Hence x and y must belong to the same atom Kk of M: That is, F maps orthogonal projections of rank one in BðHj Þ into those of BðKk Þ: Denote such k ¼ yð jÞ: Because F1 also preserves the closure of numerical range, y must be a one to one map from J onto K: By identifying BðHj Þ as a subalgebra of DN in an obvious way, we claim further that FðBðHj ÞÞDBðKyð jÞ Þ: Let A be any positive operator in BðHj Þ; then FðAÞX0 and hence FðAÞADM : If FðAÞeBðKyð jÞ Þ; then there is a j1 AJ with j1 aj such that Pyð j1 Þ FðAÞPyð j1 Þ a0; where Pyð jÞ denotes the projection from K onto Kyð jÞ : Thus, there exists a unit vector xAKyð j1 Þ and a positive number r such that Pyð j1 Þ FðAÞPyð j1 Þ Xrx#x: By Lemma 2.1, we can pick uAHj1 so that Fðu#uÞ ¼ x#x: Then W ðA ru#uÞ-ðN; 0Þa| since AABðHj Þ and uAHj1 : However, W ðFðA ru#uÞÞC½0; þNÞ by virtue of FðAÞ rx#xX0; which contradicts the assumption that F preserves the closure of numerical range. So FðAÞABðKyð jÞ Þ: By noticing that F is linear when restricted to the subspace spanned by a rank-1 projection (see Lemma 2.1), a similar argument as above shows that FðiAÞABðKyð jÞ Þ: Therefore, by the real linearity of F; for every operator T in BðHj Þ; we have FðTÞABðKyð jÞ Þ: That is, FðBðHi ÞÞDBðKyð jÞ Þ: Similarly, F1 ðBðKyð jÞ ÞÞDBðHj Þ: Hence for every jAJ; FðBðHj ÞÞ ¼ BðKyð jÞ Þ: & Note that, if L is a closed subspace of H spanned by some atoms of the nest N; then for every TADN ; we have PL T ¼ TPL ; i.e., L is a reductive subspace of T: Theorem 2.3. Let N and M be two atomic nests on H and K; respectively, and let F : DN -DM be a surjective additive map. Then F preserves the closure of numerical range if and only if there exist space decompositions H ¼ X1 "X2 and K ¼ Y1 "Y2 with Xs and Ys ðs ¼ 1; 2Þ spanned by some atoms of N and M; respectively, and there exists a unitary operator U1 : X1 -Y1 and a conjugate unitary operator U2 : X2 -Y2 such that FðTÞ ¼ U1 T1 U1 "U2 T2 U2 for every T ¼ T1 "T2 ADN with Ts ¼ PXs T ðs ¼ 1; 2Þ:
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Proof. The sufficiency is clear. Now we prove the necessity. Firstly notice that F : DN -DM is a bijective additive map since F preserves the closure of numerical range. By Lemma 2.2, there exists a 1-1 and onto map y : J-K such that for every jAJ; the restriction Fj of F to BðHj Þ is a bijective additive map preserving the closure of numerical range from BðHj Þ onto BðKyð jÞ Þ: It is implied by the proof of [1, Theorem 1] that for every jAJ; either there exists a unitary operator Uj : Hj -Kyð jÞ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þ; or there exists a conjugate unitary operator Uj : Hj -Kyð jÞ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þ: Let J1 ¼ fjAJ j there exists a unitary operator Uj ABðHj ; Kyð jÞ Þ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þg; J2 ¼ fjAJ j there exists a conjugate unitary operator Uj : Hj -Kyð jÞ such that Fj ðTj Þ ¼ Uj Tj Uj for every Tj ABðHj Þg; then J ¼ J1 ,J2 : Note that J1 or J2 may be an empty set. There is no harm in assuming that neither J1 nor J2 is empty. Put Hs ¼ "jAJs Hj and Ks ¼ "jAJs Kyð jÞ ðs ¼ 1; 2Þ; then U1 ¼ "jAJ1 Uj ABðH1 ; K1 Þ is unitary and U2 ¼ "jAJ2 Uj : H2 -K2 is conjugate unitary. Write F ¼ F1 "F2 ; where Fs ¼ Fj"jAJ BðHj Þ ðs ¼ 1; 2Þ: Set C1 ðAÞ ¼ U1 F1 ðAÞU1 for every AA"jAJ1 BðHj Þ; s then C1 ðTj Þ ¼ Tj ð8Tj ABðHj Þ; jAJ1 Þ: For each jAJ1 ; let Pj ABðHj Þ be a projection. Claim 1. If AA"jAJ1 BðHj Þ and if either AX0 or iAX0; then Pj A ¼ 0 implies Pj C1 ðAÞ ¼ C1 ðAÞPj ¼ 0: It is clear that we also have APj ¼ 0: For a sufficiently large positive number M; wðA þ MPj Þ ¼ M: It follows that wðC1 ðAÞ þ MPj Þ ¼ M: Hence for any unit vector xArngPj (the range of Pj ), j/C1 ðAÞx; xS þ MjpM:
ð2:1Þ
If AX0; then /C1 ðAÞx; xSX0; therefore inequality (2.1) entails /C1 ðAÞx; xS ¼ 0: and j/C1 ðAÞx; xS þ Mj2 ¼ If iAX0; then Reð/C1 ðAÞx; xSÞ ¼ 0 j/C1 ðAÞx; xSj2 þ M 2 ; where ReðrÞ denotes the real part of complex number r: By (2.1) again we get /C1 ðAÞx; xS ¼ 0: Thus Pj C1 ðAÞPj ¼ 0: Since C1 ðAÞX0 or iC1 ðAÞX0; one must have Pj C1 ðAÞ ¼ C1 ðAÞPj ¼ 0; as desired. Claim 2. For any projection PA"jAJ1 BðHj Þ; C1 ðPÞ ¼ P and C1 ðiPÞ ¼ iP: For any projection PA"jAJ1 BðHj Þ; P can be written into P ¼ "jAJ1 Pj with Pj being projection in BðHj Þ: Since Pj ðP Pj Þ ¼ ðP Pj ÞPj ¼ 0; by Claim 1,
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Pj ðC1 ðPÞ Pj Þ ¼ ðC1 ðPÞ Pj ÞPj ¼ 0; so C1 ðPÞPj ¼ Pj C1 ðPÞ ¼ Pj ; and therefore, PC1 ðPÞ ¼ C1 ðPÞP ¼ P: Hence, to prove C1 ðPÞ ¼ P; one need only to show that, for every jAJ1 and every xAHj ; Px ¼ 0 will imply that C1 ðPÞx ¼ 0 since N is atomic. Assume that xAHj for some jAJ1 such that Px ¼ 0; then Pðx#xÞ ¼ 0: By Claim 1 again, we get C1 ðPÞðx#xÞ ¼ 0; which implies that C1 ðPÞx ¼ 0: So C1 ðPÞ ¼ P: Similarly, one can prove C1 ðiPÞ ¼ iP ð8PA"jAJ1 BðHj ÞÞ by using Claim 1 and the fact that C1 ðiPj Þ ¼ iPj : Hence Claim 2 is true. Since "jAJ1 BðHj Þ is a von Neumann algebra and C1 is continuous in norm topology, we see that C1 ðAÞ ¼ A for all AA"jAJ1 BðHj Þ: So F1 ðAÞ ¼ U1 AU1 for all AA"jAJ1 BðHj Þ: A similar argument leads to F2 ðBÞ ¼ U2 B U2 for all BA"jAJ2 BðHj Þ: Now for every TADN ; write T ¼ T1 "T2 with Ts A"jAJs BðHj Þ ðs ¼ 1; 2Þ; we have FðTÞ ¼ U1 T1 U1 "U2 T2 U2 : The proof is completed. & Theorem 2.3 can also be restated as follows, which is sometimes convenient for applications. Theorem 2:30 . Let N and M be two atomic nests on H and K; respectively, and let F : DN -DM be a surjective additive map. Then F preserves the closure of numerical range if and only if there exist space decompositions H ¼ X1 "X2 and K ¼ Y1 "Y2 with Xs and Ys spanned by some atoms of N and M; respectively, and there exist unitary operators Us : Xs -Ys ðs ¼ 1; 2Þ such that FðTÞ ¼ U1 T1 U1 "U2 T2tr U2 for every T ¼ T1 "T2 ADN with Ts ¼ PXs T ðs ¼ 1; 2Þ; where T2tr denotes the transpose of T2 relative to a base of X2 which consists of arbitrarily fixed bases of the atoms of N contained in X2 : Remark 2.1. Let N; M and F : Alg N-Alg M be just as assumed in Lemma 2.2. Then, by Lemma 2.2 and the proof of Theorem 2.3, FjDN is a linear bijective map from DN onto DM and FjBðHj Þ has the form Ui ðÞUi or Ui ðÞ Ui ; where Ui is unitary in the former case and conjugate unitary in the latter case.
3. Additive maps preserving the closure of numerical range on maximal atomic nest algebras In this section we consider the additive maps preserving the closure of numerical range between nest algebras with maximal atomic nests. We first give a basic lemma on general atomic nest algebras which is useful in both this section and the next section. Lemma 3.1. Let N and M be two atomic nests on complex Hilbert spaces H and K; respectively, and let fHj j jAJg and fKk j kAKg be the sets of all atoms of N and M; respectively. Assume F : Alg N-Alg M is an additive surjective map and preserves
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the closure of numerical range. Then there exists a 1-1 and onto map y : J-K such that the following statements are true. (1) For any i; jAJ with ioj; and for any Tij APi Alg NPj ; PyðiÞ FðTij ÞPyð jÞ ¼ FðTij Þ if yðiÞoyð jÞ or Pyð jÞ FðTij ÞPyðiÞ ¼ FðTij Þ if yð jÞoyðiÞ: (2) For any unit vectors xi AHi and yj AHj with ioj; there exist unit vectors uyðiÞ AKyðiÞ ; vyð jÞ AKyð jÞ and complex unit aij AC such that Fðxi #xi Þ ¼ uyðiÞ #uyðiÞ ; Fðyj #yj Þ ¼ vyð jÞ #vyð jÞ and Fðxi #yj Þ ¼ aij uyðiÞ #vyð jÞ if yðiÞo yð jÞ or Fðxi #yj Þ ¼ aij vyð jÞ #uyðiÞ if yð jÞoyðiÞ: Proof. Take y : J-K just as in Lemma 2.2. (1) We claim that, for pairwise different i; j; lAJ with ioj and for any Tij APi Alg NPj ; PyðlÞ FðTij Þ ¼ 0 and FðTij ÞPyðlÞ ¼ 0; where PyðlÞ denotes the projection from K onto KyðlÞ : Assume that there exists a unit vector uAKyðlÞ such that /FðTij Þu; uSa0: By Lemma 2.1, we can find a unit vector zAHl so that Fðz#zÞ ¼ u#u: Take lAC such that jlj4jjTij jj and jl þ /FðTij Þu; uSj ¼ jlj þ j/FðTij Þu; uSj: By Lemma 2.1, Fðlz#zÞ ¼ lu#u: So wðlz#z þ Tij Þ ¼ jljojlj þ j/FðTij Þu; uSjpwðlu#u þ FðTij ÞÞ; which contradicts the assumption that F preserves the closure of numerical range. Hence /FðTij Þu; uS ¼ 0 for all uAKyðlÞ ; which implies that /FðTij Þu; vS ¼ 0 for all u; vAKyðlÞ since KyðlÞ is a complex Hilbert space. Therefore PyðlÞ FðTij ÞPyðlÞ ¼ 0 for all Tij APi Alg NPj : If there exists some Tij APi Alg NPj such that PyðlÞ FðTij Þa0; then there exists a unit vector uAKyðlÞ such that /FðTij Þv; uSa0 for some vAKyðhÞ ðhAJÞ: It is clear that loh: Take unit vector zAHl such that Fðz#zÞ ¼ u#u and take a complex number l such that jlj4jjTij jj: Then Fðlz#z þ Tij Þ has a principal submatrix of the l /FðTij Þv; uS ; where gAC: Therefore, form B ¼ 0 g wðlz#z þ Tij Þ ¼ jljowðBÞpwðFðlz#z þ Tij ÞÞ; a contradiction. Hence PyðlÞ FðTij Þ ¼ 0: FðTij ÞPyðlÞ ¼ 0 can be proved similarly. So the claim is true. For any Tij APi Alg NPj ; let FðTij Þ ¼ A: Without loss of generality, we assume that yðiÞoyð jÞ: Suppose that there exists a unit vector uAKyðiÞ such that /Au; uSa0: Then there is a unit vector zAHi such that Fðz#zÞ ¼ u#u: For any lAC satisfying jl þ /Au; uSj ¼ jlj þ j/Au; uSj; let S ¼ lz#z þ Tij : Then SS ¼ jlj2 z#z þ Tij Tij ; and hence, wðSÞ2 pjjSjj2 pjlj2 þ jjTij jj2 : Since F preserves the closure of numerical range, we have ðjlj þ j/Au; uSjÞ2 pwðlu#u þ AÞ2 pjlj2 þ jjTij jj2 ; this would lead to a contradiction that 2j/Au; uSjjlj þ j/Au; uSj2 pjjTij jj2
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for all positive numbers jlj: Therefore, we must have /Au; uS ¼ 0 for all uAKyðiÞ ; and consequently, PyðiÞ FðTij ÞPyðiÞ ¼ 0: Similarly, Pyð jÞ FðTij ÞPyð jÞ ¼ 0: Thus we have proved that PyðiÞ FðTij ÞPyð jÞ ¼ FðTij Þ: (2) For any unit vectors xi AHi and yj AHj ; let Fðxi #yj Þ ¼ A: Without loss of generality, we assume that yðiÞoyð jÞ: By (1), PyðiÞ Fðxi #yj ÞPyð jÞ ¼ Fðxi #yj Þ: Let Fðxi #xi Þ ¼ uyðiÞ #uyðiÞ and Fðyj #yj Þ ¼ vyð jÞ #vyð jÞ ; then uyðiÞ AKyðiÞ and vyð jÞ AKyð jÞ are unit vectors. Pick unit vectors uAKyðiÞ and vAKyð jÞ such that /Av; uSa0: Correspondingly, there exist unit vectors xAHi and yAHj such that Fðx#xÞ ¼ u#u and Fðy#yÞ ¼ v#v: Assume that u>uyðiÞ ; then x>xi : Take lAC with jlj41: we see that Fðlx#x þ xi #yj Þ has a principal submatrix l /Av; uS ; whose numerical radius is larger than jlj ¼ wðlx#x þ xi #yj Þ; 0 0 this is impossible. Assume that v>vyð jÞ ; then y>yj and Fðly#y þ xi #yj Þ contains a 0 /Av; uS principal submatrix ; which has the numerical radius larger than 0 l jlj ¼ wðlx#x þ xi #yj Þ; again a contradiction. Therefore, A ¼ Fðxi #yj Þ is linearly dependent to uyðiÞ #vyð jÞ : Since F preserves the closure of numerical range, there exists aij AC with jaij j ¼ 1 such that Fðxi #yj Þ ¼ aij uyðiÞ #vyð jÞ : & In the rest of this section, we always assume that N and M are two maximal atomic nests on Hilbert spaces H and K; and fHi j iAJg and fKk j kAKg be the sets of all atoms of N and M; respectively. We also suppose in the sequel that F : Alg N-Alg M is an additive surjective map which preserves the closure of numerical range and y : J-K is the 1-1 and onto map stated in Lemma 3.1. Note that F is real linear. For each iAJ; pick out a unit vector ei AHi ; then fei j iAJg is an orthonormal base of H since N is maximal atomic. Thus, for i; jAJ; ioj if and only if iaj and ei #ej AAlg N: Since M is maximal atomic, we can also choose unit vectors fk AKk such that ffk j kAKg is an orthonormal base of K: Lemma 3.2. For any j; kAJ with jok and for any xAC; Fðxej #ek Þ ¼ dðxÞFðej #ek Þ; where dðxÞ ¼ x for every xAC or dðxÞ ¼ x% for every xAC: Proof. For jAJ; replacing ej by iej in fek j kAJg; then fiej ; ek j kAJ and kajg is still a base of H: By Lemma 3.1, for j; kAJ and jok; we have Fðð1 þ iÞej #ek Þ ¼ ajk fyð jÞ #fyðkÞ þ lfyð jÞ #fyðkÞ if yð jÞoyðkÞ or Fðð1 þ iÞej #ek Þ ¼ ajk fyðkÞ #fyð jÞ þ lfyðkÞ #fyð jÞ if yðkÞoyð jÞ; where jlj ¼ 1: Since F preserves the numerical radius, pffiffiffi we must have jl þ ajk j ¼ 2; and hence l ¼ 7iajk : Now the existence of d is obvious from the real linearity of F; completing the proof. & Lemma 3.3. Assume that y : J-K is an order isomorphism (or, an anti-order isomorphism). For any i; k; jAJ with iokoj; let aij ; aik and akj be as in Lemma 3.1, then aij ¼ aik akj :
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Proof. Suppose that y : J-K is an order isomorphism. Let A ¼ ei #ek þ ek #ej þ ei #ej ; then, by Lemma 3.1, FðAÞ ¼ aik fyðiÞ #fyðkÞ þ akj fyðkÞ #fyð jÞ þ aij fyðiÞ #fyð jÞ with jaik j ¼ jakj j ¼ jaij j ¼ 1: One can check that A þ A has the smallest eigenvalue 1 and the largest eigenvalue 2: Since W ðAÞ ¼ W ðFðAÞÞ; FðAÞ þ FðAÞ also has 1 and 2 as the smallest and the largest eigenvalue, respectively. Let L be the subspace spanned by ffyðkÞ ; fyðiÞ ; fyð jÞ g; then it follows from detððFðAÞ þ FðAÞ þ IÞjL Þ ¼ 0 and jaik akj aij j ¼ 1 that aik akj aij ¼ 1; where det B denotes the determinant of matrix B: Hence aij ¼ aik akj : The proof is similar for the case that y : J-K is an anti-order isomorphism. & Lemma 3.4. Assume that y : J-K is an order isomorphism (or, an anti-order isomorphism). Then for all j; kAJ with jok; Fðiej #ek Þ ¼ iFðej #ek Þ: Proof. Let jokol and A ¼ sej #ek þ tej #el þ rek #el ; where s; t; rAC: Assume that y : J-K is an order isomorphism. By Lemmas 3.1 and 3.2, FðAÞ ¼ d1 ðsÞajk fyð jÞ #fyðkÞ þ d2 ðtÞajl fyð jÞ #fyðlÞ þ d3 ðrÞakl fyðkÞ #fyðlÞ ; here dm ðxÞ ¼ x ð8xACÞ or dm ðxÞ ¼ x% ð8xACÞ; m ¼ 1; 2; 3: This gives eight possible expressions of FðAÞ according to the following cases: Case 1: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs; t; rÞ or ðs%; t%; r%Þ: Case 2: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs%; t; rÞ or ðs; t%; r%Þ: Case 3: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs; t; r%Þ or ðs%; t%; rÞ: Case 4: ðd1 ðsÞ; d2 ðtÞ; d3 ðrÞÞ ¼ ðs; t%; rÞ or ðs%; t; r%Þ: Let L1 ¼ spanfej ; ek ; el g and L2 ¼ spanffyð jÞ ; fyðkÞ ; fyðlÞ g: Since F preserves the closure of numerical range, A þ A and FðAÞ þ FðAÞ have the same largest and smallest eigenvalues. Note that, since the coefficients of l2 are zero in both detððA þ A lÞjL1 Þ and detððFðAÞ þ FðAÞ lÞjL2 Þ; the largest and smallest roots of these two polynomials determine their third root. It turns out that detððA þ A lÞjL1 Þ ¼ detððFðAÞ þ FðAÞ lÞjL2 Þ: That is, l3 þ ðjsj2 þ jtj2 þ jrj2 Þl þ 2 Reðst%rÞ ¼ l3 þ ðjsj2 þ jtj2 þ jrj2 Þl þ 2 Reðd1 ðsÞd2 ðtÞd3 ðrÞÞ: It is clear now that Cases 1–3 cannot occur. For Case 4, i.e., d1 ðsÞ ¼ s; d2 ðtÞ ¼ t and d3 ðrÞ ¼ r or d1 ðsÞ ¼ s%; d2 ðtÞ ¼ t% and d3 ðrÞ ¼ r%; we claim that dm ðxÞ ¼ x% ðm ¼ 1; 2; 3Þ is not possible. Taking s ¼ i þ 2; t ¼ 1 i and r ¼ 1 þ i; it can be checked that the largest or smallest eigenvalues of AA and FðAÞFðAÞ are different, a contradiction. So i i dm ðxÞ ¼ x ðm ¼ 1; 2; 3Þ; this shows that, for all j; kAJ with jok; Fðiej #ek Þ ¼ iFðej #ek Þ: The case that y : J-K is an anti-order isomorphism can be treated with similarly. &
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Lemma 3.5. ð1Þ If y : J-K is an order isomorphism, then there exists a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M such that FðF Þ ¼ UFU for every F AAlgF N: ð2Þ If y : J-K is an anti-order isomorphism, then there exists a conjugate unitary operator U from H onto K satisfying UðN> Þ ¼ M such that FðF Þ ¼ UF U for every F AAlgF N: Proof. Fix some i0 AJ: Let UABðH; KÞ be defined by ( Uei ¼
aii0 fyðiÞ
if ipi0 ;
ai0 i fyðiÞ
if i4i0 :
It is clear that U is unitary. If y : J-K is an order isomorphism, then UðNÞ ¼ M: For every ei #ej AAlg N; since
Fðei #ej Þ ¼ aij fyðiÞ #fyð jÞ
8 > < aii0 aji0 fyðiÞ #fyð jÞ ¼ aii0 ai0 j fyðiÞ #fyð jÞ > : ai0 i ai0 j fyðiÞ #fyð jÞ
if ipjpi0 ; if ipi0 pj; if i0 pipj;
we have Fðei #ej Þ ¼ Uei #Uej : For any rank-1 operator x#yAAlgF N; there exists P P" i1 such that x ¼ " ipi1 xi ei and y ¼ jXi1 Zj ej ; so " X
x#y ¼
ipi1
! xi e i #
" X
! Zj e j
jXi1
¼
" X " X
xi Zj ei #ej :
ipi1 jXi1
The boundedness of F; together with Lemma 3.4, implies that X X Fðx#yÞ ¼ xi Zj Fðei #ej Þ ¼ xi Zj Uei #Uej ¼ Uðx#yÞU : ipi1 pj
ipi1 pj
So FðF Þ ¼ UFU for every F AAlgF N; i.e., (1) holds. If y : J-K is an anti-order isomorphism, then it is clear that UðN> Þ ¼ M: By Lemma 3.3, we have Fðei #ej Þ ¼ aij fyð jÞ #fyðiÞ ¼ Uej #Uei : It follows that Fðx#yÞ ¼
X ipi1 pj
xi Zj Fðei #ej Þ ¼
X
xi Zj Uej #Uei ¼ Uðx#yÞ U :
ipi1 pj
So FðF Þ ¼ UF U for all F AAlgF N; i.e., (2) holds.
&
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Let LCJ be a subset, then L is a totally ordered subset inheriting the order of J: Set PL be the projection from H onto spanfel j lALg (the closed linear span of fel j lALg). Lemma 3.6. Let LCJ be a subset and FL ¼ FjPL Alg NPL : Then FL : PL Alg NPL -PyðLÞ Alg MPyðLÞ is an additive bijective map which preserves the closure of numerical range. Proof. We first show that FL ðPL Alg NPL ÞDPyðLÞ Alg MPyðLÞ : Assume that there exists F APL Alg NPL such that FðF ÞePyðLÞ Alg MPyðLÞ ; then there is ieL or jeL such that PyðiÞ FðF ÞPyð jÞ ¼ mfyðiÞ #fyð jÞ a0 (assume that yðiÞpyð jÞ). We might as well assume that ipj: Let T ¼ aei #ei þ F ; where aAC and jaj4jjF jj: It is clear that wðTÞ ¼ jaj: Write PyðiÞ FðTÞPyðiÞ ¼ bfyðiÞ #fyðiÞ : We may assume that a is also taken so that jaþ bj ¼ jaj þ jbj; then FðTÞ has a principal matrix of the from aþb m ; where gAC; so wðFðTÞÞ4ja þ bjXjaj ¼ wðTÞ; a contradiction. Hence 0 g FL ðPL Alg NPL ÞDPyðLÞ Alg MPyðLÞ : The above argument applied to F1 yields F1 ðPyðLÞ AlgMPyðLÞ ÞDPL AlgNPL : Therefore, FL ðPL AlgNPL Þ ¼ PyðLÞ Alg MPyðLÞ : For any F APL Alg NPL ; we have F ¼ PL FPL : Let T ¼ F þ sðI PL Þ; where sAC: Then, by Remark 2.1, FðTÞ ¼ FL ðF Þ þ sðI PyðLÞ Þ: So convfW ðFL ðF ÞÞ,fsgg ¼ W ðFðTÞÞ ¼ W ðTÞ ¼ convfW ðF Þ,fsgg: Taking sAW ðF Þ; one gets W ðFL ðF ÞÞDW ðF Þ; picking sAW ðFL ðF ÞÞ; one obtains W ðF ÞDW ðFL ðF ÞÞ: So for every F APL Alg NPL ; we always have W ðFL ðF ÞÞ ¼ W ðF Þ: & Let Eij denote the standard matrix units of Mn ðCÞ ði; j ¼ 1; y; nÞ and Tn be the upper triangular matrix algebras in Mn ðCÞ: The proof of the following lemma can be found in [11]. Lemma 3.7. Let f : Tn -Tn be an additive map which preserves the numerical range. Then there exists a 1-1 map W : f1; 2; y; ng-f1; 2; y; ng which satisfies fðEii Þ ¼ EWðiÞWðiÞ ð1pipnÞ; and there exists jAf1; 2; y; ng such that Wð jÞ ¼ 1 and one of the following holds: (1) Wð jÞoWð j þ 1Þo?oWðn 1ÞoWðnÞoWð1ÞoWð2Þo?oWð j 1Þ: (2) Wð jÞoWð j 1Þo?oWð2ÞoWð1ÞoWðnÞoWðn 1Þo?oWð j þ 1Þ: Let Js ðs ¼ 1; 2Þ be totally ordered sets, we denote the order sum of J1 and J2 by J ¼ J1 þ J2 ; which is a totally ordered set J ¼ J1 ,J2 (regard as J1 -J2 ¼ |), in
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which ioj means ioj if i; jAJs ; or if iAJ1 and jAJ2 : Note that, by the definition, J1 þ J2 aJ2 þ J1 in general. Lemma 3.8. isomorphism, ð1Þ There y : Js -yðJs Þ ð2Þ There y : Js -yðJs Þ
If y : J-K is neither an order isomorphism nor an anti-order then one of the following holds: exist J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ1 Þ þ yðJ2 Þ and is an anti-order isomorphism, s ¼ 1; 2: exist J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ2 Þ þ yðJ1 Þ and is an order isomorphism, s ¼ 1; 2:
Proof. Assume that y : J-K is neither an order isomorphism nor an anti-order isomorphism, then there exist i1 ; i2 ; i3 AJ with i1 oi2 oi3 such that one of the following holds: (i) (ii) (iii) (iv)
yði1 Þoyði3 Þoyði2 Þ; yði2 Þoyði3 Þoyði1 Þ; yði2 Þoyði1 Þoyði3 Þ; yði3 Þoyði1 Þoyði2 Þ:
Suppose that (i) holds. For any j1 ; j2 ; j3 AJ with j1 oj2 oj3 ; it is easily checked that fi1 ; i2 ; i3 ; j1 ; j2 ; j3 g has totally the following 20 possible orders: ð11Þ ð21Þ ð31Þ ð41Þ ð51Þ ð61Þ ð71Þ ð81Þ ð91Þ ð101Þ ð111Þ ð121Þ ð131Þ ð141Þ ð151Þ ð161Þ ð171Þ ð181Þ ð191Þ ð201Þ
i1 oi2 oi3 pj1 oj2 oj3 ; i1 oi2 pj1 oi3 pj2 oj3 ; i1 pj1 oi2 oi3 pj2 oj3 ; i1 oi2 pj1 oj2 pi3 oj3 ; i1 pj1 oj2 pi2 oi3 pj3 ; i1 pj1 oi2 pj2 oi3 pj3 ; i1 pj1 oj2 oj3 pi2 oi3 ; i1 pj1 oj2 pi2 oj3 pi3 ; i1 pj1 oi2 pj2 oj3 pi3 ; i1 oi2 pj1 oj2 oj3 pi3 ; j1 pi1 oi2 pj2 oj3 pi3 ; j1 oj2 pi1 oi2 pj3 oi3 ; j1 oj2 pi1 oj3 pi2 oi3 ; j1 oj2 oj3 pi1 oi2 oi3 ; j1 pi1 oj2 pi2 oj3 pi3 ; j1 pi1 oj2 oj3 pi2 oi3 ; j1 pi1 oi2 oi3 pj2 oj3 ; j1 pi1 oi2 pj2 oi3 pj3 ; j1 pi1 oj2 pi2 oi3 pj3 ; j1 oj2 pi1 oi2 oi3 pj3 :
Let L1 ¼ fi1 ; i2 ; i3 ; j1 ; j2 ; j3 g and L2 ¼ yðL1 Þ: By Lemma 3.6, FjPL
1
Alg NPL1
: PL1 Alg NPL1 -PL2 Alg NPL2
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preserves the numerical range. Using Lemma 3.7, it is easily checked that, in each of 20 possible cases listed above, fyð j1 Þ; yð j2 Þ; yð j3 Þg can only have one of the order relations (i), (iii) and the anti-order yð j3 Þoyð j2 Þoyð j1 Þ: Similarly, that (iii) holds also implies that fyð j1 Þ; yð j2 Þ; yð j3 Þg can only have one of the order relations (i), (iii) and the anti-order. Now we consider the following three cases. Case 1: (i) occurs but (iii) does not. Take i0 such that i0 oi1 oi2 ; then yði0 Þoyði2 Þoyði1 Þ; which contradicts yði1 Þoyði2 Þ: So there is no i0 such that i0 oi1 : Hence, i1 ¼ min J and yði1 Þ ¼ min K: Therefore, for any j2 ; j3 ; y; jn with i1 oj2 oj3 o?ojn ; we have yði1 Þoyð jn Þoyð jn1 Þo?oyð j2 Þ: Put J ¼ J1 þ J2 and K ¼ K1 þ K2 ; where J1 ¼ fi1 g and K1 ¼ fyði1 Þg; then y : J2 -K2 is an anti-order isomorphism. Case 2: (iii) occurs but (i) does not. A similar argument just as in Case 1 implies that there exists j1 AJ such that j1 ¼ max J and yð j1 Þ ¼ max K: Let J ¼ J1 þ J2 and K ¼ K1 þ K2 ; where J2 ¼ fj1 g and K2 ¼ fyð j1 Þg; we have y : J1 -K1 is an anti-order isomorphism. Case 3: Both (i) and (iii) occur. Set J1 ¼ fiAJ j there are j; k such that iojok and yðiÞoyðkÞoyð jÞg; J2 ¼ flAJ j there are j; k such that jokol and yðkÞoyð jÞoyðlÞg: Note that iAJ1 and hoi imply hAJ1 ; lAJ2 and lom imply mAJ2 : Notice also that, if h; iAJ1 and hoi; then yðhÞ4yðiÞ; if l; mAJ2 and lom; then yðlÞ4yðmÞ: By Cases 1 and 2, J1 a| and J2 a|: If iAJ1 -J2 ; then there are gohoiokol such that yðhÞoyðgÞoyðiÞoyðlÞoyðkÞ; this is impossible by Lemmas 3.6 and 3.7. So J1 -J2 ¼ |; and therefore, J1 oJ2 ; i.e., for any iAJ1 and lAJ2 ; we have iol: We assert that for any iAJ1 and lAJ2 ; yðiÞoyðlÞ: Otherwise, assume that yðlÞoyðiÞ: Take j; kAJ such that jokol; then yðkÞoyð jÞoyðlÞ: It is clear that fi; j; k; lg have three possible order relations: ð10 Þ ipjokol; ð20 Þ joipkol; ð30 Þ jokoiol: However, by Lemmas 3.6 and 3.7, ð10 Þ would imply yðkÞoyð jÞpyðiÞoyðlÞoyðiÞ and ð20 Þ would imply yðkÞpyðiÞoyð jÞoyðlÞoyðiÞ; contradictions. ð30 Þ would imply that yðiÞoyðkÞoyð jÞoyðlÞoyðiÞ or yðkÞoyð jÞoyðlÞoyðiÞ: The former is impossible and for the latter, we would have iAJ2 ; a contradiction. So the assertion holds. If J ¼ J1 ,J2 ; then the proof is completed. Assume that JaJ1 ,J2 ; then J\ðJ1 ,J2 Þa|: For any jAJ\ðJ1 ,J2 Þ; we have J1 ojoJ2 : Since H is infinite dimensional, there is at least one among J1 ; J3 ¼ J\ðJ1 ,J2 Þ and J2 which is an % ¼ J\ðJ ,J ÞX3; where S % denotes the cardinal of the set infinite set. Note that if J 3 1 2 S; we may take i; j; kAJ3 such that iokoj: Let hAJ1 ; lAJ2 ; then hoiojokol: Since yðhÞoyðlÞ; we have yðkÞoyð jÞoyðiÞoyðhÞoyðlÞ or yðhÞoyðlÞ oyðkÞoyð jÞoyðiÞ; so y is anti-order when restricted to J3 and either
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yðJ3 ÞoyðJ1 Þ or yðJ2 ÞoyðJ3 Þ: If yðJ3 ÞoyðJ1 Þ occurs, let J01 ¼ J1 þ J3 ; J02 ¼ J2 ; if yðJ2 ÞoyðJ3 Þ occurs, let J01 ¼ J1 ; J02 ¼ J3 þ J2 : Then J ¼ J01 þ J02 ; yðJ01 ÞoyðJ02 Þ and y is anti-order on J01 and J02 : If 1pJ% 3 p2; then at least one of J1 and J2 is an infinite set. Without loss of generality, we assume that J1 is an infinite set. For any jAJ3 ; pick i1 ; i2 AJ1 with i1 oi2 and lAJ2 ; then i1 oi2 ojol: Due to yðis ÞoyðlÞ ðs ¼ 1; 2Þ; we have either (a) yði2 Þoyði1 ÞoyðlÞoyð jÞ; or (b) yð jÞoyði2 Þoyði1 ÞoyðlÞ: We claim that (a) and (b) cannot occur simultaneously. Indeed, if (a) holds for i1 oi2 ojol; and (b) holds for h1 oh2 ojom; then yði2 Þoyði1 ÞoyðlÞo yð jÞoyði2 Þoyði1 ÞoyðlÞ; and therefore, i1 oi2 omoh1 oh2 ojol or h1 oh2 ojoloi1 oi2 om; which contradicts to the assumption h1 oh2 ojom or i1 oi2 ojol: Hence yðJ3 ÞoyðJ1 Þ or yðJ3 Þ4yðJ2 Þ: If J3 ¼ fj; kg with jok; then it can be seen from yðJ3 Þ4yðJ2 Þ or yðJ3 ÞoyðJ1 Þ that yðkÞoyð jÞ: So y is still antiorder on J3 if 1pJ% 3 p2: In the case yðJ3 ÞoyðJ1 Þ; let J01 ¼ J1 þ J3 ; J02 ¼ J2 ; in the case yðJ3 Þ4yðJ2 Þ; let J01 ¼ J1 ; J02 ¼ J3 þ J2 ; then J ¼ J01 þ J02 ; as desired. All in all, if (i) or (iii) occurs, then there exists a decomposition J ¼ J1 þ J2 such that yðJ1 ÞoyðJ2 Þ and, when restricted to Js ; y : Js -yðJs Þ is an anti-order isomorphism ðs ¼ 1; 2Þ: If (ii) or (iv) occurs, one can similarly prove that there exists a decomposition J ¼ J1 þ J2 such that yðJ1 Þ4yðJ2 Þ and y : Js -yðJs Þ is an order isomorphism (s ¼ 1; 2). & The following examples show that there are many additive maps on nest algebras which preserve numerical ranges but are neither linear nor conjugate linear. Let Ns be a nest on Hilbert space Hs ; s ¼ 1; 2; and let N ¼ N1 þ N2 consist of N1 "0 and H1 "N2 ; where Ns ANs ðs ¼ 1; 2Þ: It is clear that N is a nest on H ¼ H1 "H2 : Example 3.1. Let N ¼ N1 þ N2 and M ¼ M2 þ M1 be two nests on Hilbert spaces H ¼ H1 "H2 and K ¼ K2 "K1 ; respectively. Assume that unitary operators Us ABðHs ; Ks Þ satisfy Us ðNs Þ ¼ Ms ðs¼ 1; 2Þ: For TAAlg N; with respect to the T1 T12 above decompositions, one has T ¼ with Ts AAlg Ns ðs ¼ 1; 2Þ and 0 T2 U2 T2 U2 U2 T12 U1 T12 ABðH2 ; H1 Þ: Define G : Alg N-Alg M by GðTÞ ¼ : It 0 U1 T1 U1 is clear GðTÞAAlg M: Let x ¼ x1 "x2 AH with jjxjj ¼ 1 and let y ¼ aU2 x2 "U1 x1 ; where ( a¼
1
if /T12 x2 ; x1 S ¼ 0;
/x1 ;T12 x2 S /T12 x2 ;x1 S
if /T12 x2 ; x1 Sa0:
ð3:1Þ
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Then /GðTÞy; yS ¼ /aU2 T2 x2 ; aU2 x2 S þ /U1 T1 x1 ; U1 T1 x1 S þ /U2 T12 x1 ; aU2 x2 S
¼ /T2 x2 ; x2 S þ /T1 x1 ; x1 S þ /T12 x2 ; x1 S ¼ /Tx; xS: We conclude that W ðGðTÞÞ+W ðTÞ: On the other hand, for any unit vector y ¼ y2 "y1 AK ¼ K2 "K1 ; let x ¼ bU1 y1 "U2 y2 AH ¼ H1 "H2 ; where ( b¼
1
if /T12 U2 y2 ; U1 y1 S ¼ 0;
/T12 U2 y2 ;U1 y1 S /U1 y1 ;T12 U2 y2 S
if /T12 U2 y2 ; U1 y1 Sa0:
ð3:2Þ
A computation shows that % /Tx; xS ¼ /T1 U1 y1 ; U1 y1 S þ /T2 U2 y2 ; U2 y2 S þ b/T 12 U2 y2 ; U1 y1 S U1 y1 ; y2 S ¼ /U1 T1 U1 y1 ; y1 S þ /U2 T2 U2 y2 ; y2 S þ /U2 T12
¼ /GðTÞy; yS: This implies that W ðGðTÞÞDW ðTÞ: So W ðGðTÞÞ ¼ W ðTÞ ð8TAAlg NÞ: Obviously, G is a surjective additive map, but G is neither linear nor conjugate linear. Example 3.2. Let N ¼ N1 þ N2 and M ¼ M1 þ M2 be two nests on Hilbert spaces H ¼ H1 "H2 and K ¼ K1 "K2 ; respectively. Assume that conjugate unitary operators Us : Hs -Ks satisfy Us ðN> For TAAlg N; s Þ ¼ Ms ðs ¼ 1; 2Þ: T1 T12 write T ¼ with Ts AAlg Ns ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: Let 0 T2 U1 T1 U1 U1 T12 U2 D : Alg N-Alg M be defined by DðTÞ ¼ : Clearly DðTÞA 0 U2 T2 U2 Alg M and D is additive and surjective but is neither linear nor conjugate linear. We claim that D preserves the numerical range. For any x ¼ x1 "x2 AH with jjxjj ¼ 1: Put y ¼ aU1 x1 "U2 x2 ; where a is taken as in (3.1). Then jjyjj ¼ 1 and /DðTÞy; yS ¼ /aU1 T1 x1 ; aU1 x1 S þ /U2 T2 x2 ; U2 x2 S þ /U1 T12 x2 ; aU1 x1 S ¼ /x1 ; T1 x1 S þ /x2 ; T2 x2 S þ /T12 x2 ; x1 S ¼ /Tx; xS: So W ðTÞDW ðDðTÞÞ: For any unit vector y ¼ y1 "y2 ; put x ¼ bU1 y1 "U2 y2 ; where b is taken as in (3.2). It is easily seen that /DðTÞy; yS ¼ /Tx; xS; and hence W ðDðTÞÞDW ðTÞ: Therefore D preserves the numerical range.
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Note that G and D in the above examples are weakly continuous. To our surprise, from the main result Theorem 4.1 in next section, for the weakly continuous and surjective additive maps between atomic nest algebras which preserve the closure of numerical range and which are neither linear nor conjugate linear, Examples 3.1 and 3.2 are only possible ones. We first prove this in the next result for the case of maximal atomic nest algebras. Let N be a nest on H: For any H1 AN; set N1 ¼ fN-H1 j NANg and N2 ¼ fN-H2 j NANg; where H2 ¼ H1> : Then Ns is a nest on Hs ðs ¼ 1; 2Þ and N ¼ N1 þ N2 : We say this space decomposition and the corresponding nest decomposition by H1 : Thus for every TAAlg N; T can be written as is determined T1 T12 T¼ ; where Ts AAlg Ns ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: 0 T2 Theorem 3.9. Let N and M be two maximal atomic nests on Hilbert spaces H and K; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective additive map. Then F preserves the closure of numerical range if and only if one of the following holds: (1) There exists a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M such that FðTÞ ¼ UTU for every TAAlg N: (2) There exists a conjugate unitary operator U from H onto K satisfying UðN> Þ ¼ M such that FðTÞ ¼ UT U for every TAAlg N: (3) There exist H1 AN; K1 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively; there exist conjugate unitary operators Us : Hs -Ks satisfying Us ðN> s Þ ¼ Ms ðs ¼ 1; 2Þ; such that ! ! T1 T12 U1 T1 U1 U1 T12 U2 ; 8T ¼ AAlg N: FðTÞ ¼ 0 U2 T2 U2 0 T2 (4) There exist H1 AN; K2 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 ; respectively; there exist unitary operators Us : Hs -Ks satisfying Us ðNs Þ ¼ Ms ðs ¼ 1; 2Þ; such that ! ! U2 T2 U2 U2 T12 U1 T1 T12 FðTÞ ¼ ; 8T ¼ AAlg N: 0 U1 T1 U1 0 T2 Proof. Let fHi j iAJg and fKj j jAKg be the set of all atoms of N and M; respectively. By Lemma 2.2, y : J-K is one to one and onto. If y : J-K is an order isomorphism or an anti-order isomorphism, then by Lemma 3.5 and the weak continuity of F; we have that the result (1) or (2) holds.
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Now suppose that y : J-K is neither an order isomorphism nor an anti-order isomorphism. By Lemma 3.8, one of the following holds: (i) There exist y : Js -yðJs Þ (ii) There exist y : Js -yðJs Þ
J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ1 Þ þ yðJ2 Þ and is an anti-order isomorphism for each s ðs ¼ 1; 2Þ: J1 and J2 such that J ¼ J1 þ J2 ; K ¼ yðJ2 Þ þ yðJ1 Þ and is an order isomorphism for each s ðs ¼ 1; 2Þ:
Assume that (i) holds. Let K1 ¼ yðJ1 Þ; K2 ¼ yðJ2 Þ and let H1 ¼ spanfej j jAJ1 gAN; K1 ¼ spanffk j kAK1 gAM: Then H1 and K1 determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively. It is clear that Ns and Ms are maximal atomic nests on Hs and Ks ðs ¼ 1; 2Þ; respectively. By Lemma 3.1(2), if i; jAJs ðs ¼ 1; 2Þ with ioj; then Fðei #ej Þ ¼ aij fyð jÞ #fyðiÞ ; if iAJ1 ; jAJ2 ; then Fðei #ej Þ ¼ aij fyðiÞ #fyð jÞ ; where aij AC and jaij j ¼ 1; and for any finite rank operator F ABðH2 ; H1 Þ; FðF ÞABðK2 ; K1 Þ is a finite rank operator. Set Fs ¼ FjAlg Ns ; s ¼ 1; 2: By Lemma 3.6, Fs : Alg Ns -Alg Ms are additive bijective maps which preserve the closure of numerical range. Hence by Lemma 3.5(2), for each s ðs ¼ 1; 2Þ; there exists a conjugate unitary operator Us from Hs onto Ks satisfying Us ðN> s Þ ¼ Ms such that Fs ðTs Þ ¼ Us Ts Us for every Ts AAlg Ns : Let LCJ be an arbitrary finite subset and PL denote the projection from P H onto HL ¼ " L2 ¼ L-J2 ; then L ¼ L1 þ L2 iAL Hi : Set L1 ¼ L-J1 and T1 T12 and yðLÞ ¼ yðL1 Þ þ yðL2 Þ: For each T ¼ AAlg N; PL TPL ¼ 0 T2 PL1 T1 PL1 PL1 T12 PL2 : 0 PL2 T2 PL2 Note that FL ¼ FjPL Alg NPL can be regarded as an additive map from the upper triangular matrix algebra Tn onto Tn which preserves the numerical range, where % By using the result [11, Theorem 11] for Tn ; it is seen that the strictly upper n ¼ L: triangular parts of FL ðPL TPL Þ are uniquely determined by the map y : L-yðLÞ: T1 T12 Meanwhile Example 3.2 implies that, for each T ¼ ; if DðTÞ ¼ 0 T2 U1 T1 U1 U1 T12 U2 ; then W ðDðTÞÞ ¼ W ðTÞ: So, by the uniqueness of the 0 U2 T2 U2 expression of FL we must have ! U1 PL1 T1 PL1 U1 U1 PL1 T12 PL2 U2 FðPL TPL Þ ¼ : 0 U2 PL2 T2 PL2 U2 Now the weak continuity of F yields that ! U1 T1 U1 U1 T12 U2 FðTÞ ¼ ; 0 U2 T2 U2
8T ¼
T1
T12
0
T2
! AAlg N:
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If (ii) occurs, let H1 ¼ spanfej j jAJ1 gAN and K2 ¼ spanffk j kAK2 gAM: Then H1 and K2 determine, respectively, the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 : Obviously, Ns and Ms are maximal atomic nests on Hs and Ks (s ¼ 1; 2), respectively. Now, applying Lemma 3.5(1), Example 3.1, and a similar argument just as in case (i), we see that F has the form (4) stated in the theorem. The proof is completed. & By Examples 3.1, 3.2 and Theorem 3.9, the following corollary is immediate. Corollary 3.10. Let N and M be two maximal atomic nests on Hilbert spaces H and K; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective additive map. Then the following statements are equivalent: (1) F preserves the numerical range. (2) F preserves the closure of numerical range. (3) F has one of the four forms stated in Theorem 3.9. Compared with Theorem 2.3, we conjecture that the weak continuity assumption for F in Theorem 3.9 may be superfluous. However, we are not able to omit this assumption at present. Our next result says that, for a class of special nests, the conjecture is true. Let o denote the order type of the set N of natural numbers, and Wn o be the order type of N: Let fej gN j¼1 be a base of H; and let Nn ¼ j¼1 fej g; then N ¼ ff0g; Nn ðnX1Þ; Hg is a maximal atomic nest on H of the order type o þ 1; and N> has order type 1 þ o : All maximal atomic nests of order type o þ 1 or 1 þ o appear in this way. Theorem 3.11. Let N be a maximal atomic nest on H of the order type o þ 1 or 1 þ o : Assume that F : Alg N-Alg N is an additive surjective map. Then F preserves the closure of numerical range if and only if there exists a unitary operator UABðHÞ satisfying UðNÞ ¼ N such that FðTÞ ¼ UTU for every TAAlg N: Proof. Assume that N has order type 1 þ o : Let CðT tr Þ ¼ FðTÞtr for each TAAlg N; where T tr is the transpose of T with respect to the orthonormal base consisting of the unit vectors in atoms of N: Then C : Alg N> -Alg N> is additive and preserves the closure of numerical range. Therefore, it suffices to check the case that N has order type o þ 1: Suppose that N has order type o þ 1 and take an orthonormal base fej gjAJ of H consists of unit vectors from the atoms of N: Then J ¼ N and it is obvious that the map y : J-J stated in Lemma 3.1 is identity. By Lemma 3.5, there is a unitary operator UABðHÞ satisfying UðNÞ ¼ N such that FðF Þ ¼ UFU for all F AAlgF N: Let CðTÞ ¼ U FðTÞU for each TAAlg N: Then C : Alg N-Alg N is an additive surjective map preserving the closure of numerical range and CðF Þ ¼ F for every F AAlgF N: We claim that CðTÞ ¼ T for each TAAlg N; and therefore, the result holds.
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For any TAAlg N; let CðTÞ ¼ S: If SaT; then there are some n and m in N with mXn such that /Sem ; en Sa/Tem ; en S: Without loss of generality, we assume that n; m are the smallest numbers satisfying the above inequality. Write T ¼ PN PN Pn PN PN PN j¼1 i¼j tji ej #ei and S ¼ j¼1 i¼j sji ej #ei : Take F0 ¼ j¼1 i¼j tji ej #ei : Then F0 þ T ¼
N N X X j¼nþ1
F0 þ S ¼
n N X X j¼1
i¼j
tji ej #ei ;
i¼j
ðsji tji Þej #ei þ
N N X X j¼nþ1
sji ej #ei :
i¼j
Let a ¼ smn tmn ; then aa0: Choose mAC such that jmj4jjT þ F0 jj and jm þ aj ¼ jmj þ jaj: Let F ¼ men #en ; then wðT þ F0 þ F Þ ¼ jmj; but we always have wðS þ F0 þ F Þ4jmj; which contradicts to the fact that W ðT þ F0 þ F Þ ¼ W ðS þ F0 þ F Þ: Hence S ¼ T: & Theorem 3.11 implies that only the case (1) in Theorem 3.9 occurs when both N and M have the order type o þ 1 (or, 1 þ o ). In the rest of this section, we will further illustrate some examples to show that almost all possible combinations of four forms (1)–(4) stated in Theorem 3.9 may occur. (a) If N and M have the order type o þ 1 and 1 þ o ; respectively, then only the case (2) in Theorem 3.9 occurs. (b) If N and M have the order type o þ 1 þ n and 1 þ o ð¼ 1 þ o þ nÞ; respectively, then only the case (3) in Theorem 3.9 occurs. (c) If N and M have the order type o þ 1 þ n and o þ 1 ð¼ n þ o þ 1Þ; respectively, then only the case (4) in Theorem 3.9 occurs. (d) If both N and M have the order type 1 þ o þ o þ 1; then the cases (1) and (2) in Theorem 3.9 occur, but (3) and (4) do not occur. (e) If both N and M have the order type n þ o þ o þ 1; then the cases (1) and (3) in Theorem 3.9 occur, but (2) and (4) do not occur. (f) If both N and M have the order type o þ o þ 1; then the cases (1) and (4) in Theorem 3.9 occur, but (2) and (3) do not occur. (g) If N and M have the order type o þ o þ 1 and 1 þ o þ o ; respectively, then the cases (2) and (3) in Theorem 3.9 occur, but (1) and (4) do not occur. (h) If N and M have the order type n þ o þ o þ 1 and 1 þ o þ o þ n; respectively, then the cases (2) and (4) in Theorem 3.9 occur, but (1) and (3) do not occur. (i) If N and M have the order type ðo þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o Þ þ ð1 þ o þ o þ 1Þ and ð1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1Þ þ ðo þ 1 þ o Þ; respectively, then the cases (3) and (4) in Theorem 3.9 occur, but (1) and (2) do not occur.
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( j) If both N and M have the order type ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ; then the cases (1), (3) and (4) in Theorem 3.9 occur, but (2) does not occur. (k) If N and M have the order type ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ and ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ þ ðo þ 1 þ o Þ þ ð1 þ o þ o þ 1Þ; respectively, then the cases (2)–(4) in Theorem 3.9 occur, but (1) does not occur. (l) If both N and M have the order type o þ 1 þ o ; then all cases (1)–(4) in Theorem 3.9 occur. We remark that if the cases (1)–(3) hold in Theorem 3.9, then (4) holds, too. Let J and K be the totally ordered index sets in the proof of Theorem 3.9. Then there exists an order isomorphism y : J-K; an anti-order isomorphism t : J-K; and there exist decompositions J ¼ J1 þ J2 and K ¼ K1 þ K2 ; and anti-order isomorphisms ds : Js -Ks ðs ¼ 1; 2Þ: Let Zs ¼ y 3 t1 3 ds ðs ¼ 1; 2Þ; it is clear that Zs : Js -Ks are order isomorphisms and K ¼ Z2 ðJ2 Þ þ Z1 ðJ1 Þ: This implies that form (4) occurs, too. Similarly forms (1), (2) and (4) together will imply (3) in Theorem 3.9. Hence fð1Þ; ð2Þ; ð3Þg3fð1Þ; ð2Þ; ð4Þg3fð1Þ; ð2Þ; ð3Þ; ð4Þg: So the above examples exhaust all possible combinations.
4. Additive maps preserving the closure of numerical range on atomic nest algebras The discussions in Sections 2 and 3 make it possible to characterize further the additive maps on atomic nest algebras which preserve the closure of the numerical ranges. Theorem 4.1. Let N and M be atomic nests on complex Hilbert spaces H and K with atoms fHj j jAJg and fKk j kAKg; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective additive map. Then F preserves the closure of numerical range if and only if one of the following statements is true: (1) There is a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M such that FðTÞ ¼ UTU for all TAAlg N: (2) There is a conjugate unitary operator U : H-K satisfying UðN> Þ ¼ M such that FðTÞ ¼ UT U for all TAAlg N: (3) There exist H1 AN; K1 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively; there exist conjugate unitary operators Us : Hs -Ks satisfying Us ðN> s Þ ¼ Ms ðs ¼ 1; 2Þ; such that FðTÞ ¼
U1 T1 U1
U1 T12 U2
0
U2 T2 U2
! ;
8T ¼
T1
T12
0
T2
! AAlg N:
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(4) There exist H1 AN; K2 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 ; respectively; there exist unitary operators Us : Hs -Ks satisfying Us ðNs Þ ¼ Ms ðs ¼ 1; 2Þ; such that ! ! U2 T2 U2 U2 T12 U1 T1 T12 FðTÞ ¼ ; 8T ¼ AAlg N: 0 U1 T1 U1 0 T2 Proof. By Examples 3.1 and 3.2, we need only to check the ‘‘only if’’ part. Assume that F preserves the closure of numerical range. It follows from Remark 2.1 and Theorem 2:30 that there exists a one–one and onto map y : J-K and the restriction of F to DN is a linear map from DN onto DM such that
F " Tj jAJ
¼ " Uyð jÞ Tjw Uyð jÞ ; jAJ
where Uyð jÞ is a unitary from Hj onto Kyð jÞ ; Tjw ¼ Tj or Tjtr : Hereafter, without loss of generality, we may regard Kyð jÞ as Hj : Hence Uyð jÞ ABðHj Þ and K ¼ H: It is certainly possible that for every jAJ we can take an orthonormal base feði;jÞ j iAJj g of Hj so that Jj is a totally ordered index set and there exists an anti-order isomorphism tj from Jj onto itself satisfying that t2j is the identity idj on Jj : Let the transpose Tjtr of Tj be taken with respect to such orthonormal base. Fix an anti-order isomorphism tj : Jj -Jj satisfying t2j ¼ idj and let Ej be the unitary operator on Hj determined by Ej eði;jÞ ¼ eðtj ðiÞ;jÞ : It is obvious that Ej2 ¼ Ij (the identity on Hj ). Put U ¼ "kAK Uk Dy1 ðkÞ ; where Dy1 ðkÞ ¼ Iy1 ðkÞ if Tyw1 ðkÞ ¼ Ty1 ðkÞ ; Dy1 ðkÞ ¼ Ey1 ðkÞ if Tyw1 ðkÞ ¼ Tytr1 ðkÞ : Then U is a unitary operator from H onto K satisfying UðMÞ ¼ M by noting that Kk ¼ Hy1 ðkÞ : Denote Tjf ¼ Ej Tjtr Ej and let CðÞ ¼ U FðÞU: It is clear that C is additive, surjective and preserves the closure of numerical range from z Alg N onto Alg M: Moreover, for T ¼ "jAJ Tj ADN ; CðTÞ ¼ "kAK Ty1 ðkÞ ; where z Ty1 ðkÞ ¼ Ty1 ðkÞ or Tyf1 ðkÞ : # ¼ S fði; jÞ j iAJ g be the totally ordered set with ði; jÞoðl; mÞ if and only Let J j jAJ # is defined similarly. Let N0 and M0 be the maximal if jom or iol when j ¼ m: K atomic nests generated by the bases # feði;jÞ j ði; jÞAJg
ð4:1Þ
# feðl;kÞ j ðl; kÞAKg;
ð4:2Þ
and
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respectively. Obviously, Alg N0 DAlg N; Alg M0 DAlg M and CðDN -Alg N0 Þ ¼ DM -Alg M0 ;
CðLN Þ ¼ LM ;
here LN ¼ fTADN j T is strictly lower triangular with respect to the maximal atomic base in ð4:1Þg and LM is one with respect to (4.2). By Lemma 3.1 and the weak continuity of C; it is also clear that C maps Alg N0 onto Alg M0 : Thus it follows from Theorem 3.9 that C takes one of the following forms when restricted to Alg N0 : (i) There is a unitary operator V ABðH; KÞ satisfying V ðN0 Þ ¼ M0 such that CðTÞ ¼ VTV for all TAAlg N0 : (ii) There is a conjugate unitary operator V : H-K satisfying V ððN0 Þ> Þ ¼ M0 such that CðTÞ ¼ VT V for all TAAlg N0 : (iii) There exist H1 AN0 ; K1 AM0 which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N0 ¼ N01 þ N02 ; M0 ¼ M01 þ M02 ; respectively; there exist conjugate unitary operators Vs : Hs -Ks satisfying Vs ððN0s Þ> Þ ¼ M0s ðs ¼ 1; 2Þ; such that ! ! V1 T1 V1 V1 T12 V2 T1 T12 CðTÞ ¼ ; 8T ¼ AAlg N0 ; 0 V2 T2 V2 0 T2 where Ts AAlg N0s ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: (iv) There exist H1 AN0 ; K2 AM0 which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N0 ¼ N01 þ N02 ; M0 ¼ M02 þ M01 ; respectively; there exist unitary operators Vs ABðHs ; Ks Þ satisfying Vs ðN0s Þ ¼ M0s ðs ¼ 1; 2Þ; such that ! ! V2 T2 V2 V2 T12 T1 T12 V1 CðTÞ ¼ ; 8T ¼ AAlg N0 ; 0 V1 T1 V1 0 T2 where Ts AAlg N0s ðs ¼ 1; 2Þ and T12 ABðH2 ; H1 Þ: To see how to go back to the atomic nest algebras, say (iii) occurs. Obviously, one must request that H1 AN and K1 AM: Thus H1 and K1 determine nest decompositions N ¼ N1 þ N2 and M ¼ M1 þ M2 ; respectively. Recall that
C " Tj jAJ
¼ " Tyf1 ðkÞ ¼ " Ey1 ðkÞ Jy1 ðkÞ Ty1 ðkÞ Jy1 ðkÞ Ey1 ðkÞ ; kAK
kAK
ð4:3Þ
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where Jj is a conjugate unitary operator on Hj defined by 0 1 X X Jj @ xeði;jÞ A ¼ x% eði;jÞ : ði;jÞAJj
ði;jÞAJj
Note that Vs ðHj Þ ¼ Kyð jÞ for each jAJs ; s ¼ 1; 2: Let Vk ¼ Vs jH 1 y
ðkÞ
if kAKs : Then,
for any T ¼ T1 "T2 ¼ ð"jAJ1 Tj Þ"ð"jAJ2 Tj Þ ¼ "jAJ Tj ADN -Alg N0 ; we have ! V1 T1 V1 0 CðTÞ ¼ ð4:4Þ ¼ " Vk Ty1 ðkÞ Vk : 0 V2 T2 V2 kAK Comparing (4.3) with (4.4), one has Ey1 ðkÞ Jy1 ðkÞ Ty1 ðkÞ Jy1 ðkÞ Ey1 ðkÞ ¼ Vk Ty1 ðkÞ Vk and hence, Ey1 ðkÞ Jy1 ðkÞ Ty1 ðkÞ Jy1 ðkÞ Ey1 ðkÞ ¼ Vk Ty1 ðkÞ Vk for every kAK: Now it is easily checked that the formula in (iii) also holds when C is restricted to LN : Since FðÞ ¼ UCðÞU and UðMÞ ¼ M; we see that F takes form (3) stated in the theorem. The other cases are dealt similarly. The proof is complete. & For the linear case, we have the following result which in particular generalizes [2, Theorem 2.5]. Theorem 4.2. Let N and M be atomic nests on complex Hilbert spaces H and K; respectively. Suppose that F : Alg N-Alg M is a weakly continuous and surjective linear map. Then F preserves the closure of numerical range if and only if one of the following holds: (1) There is a unitary operator U : H-K satisfying UðNÞ ¼ M such that FðTÞ ¼ UTU for all TAAlg N: (2) There is a conjugate unitary operator U : H-K satisfying UðN> Þ ¼ M such that FðTÞ ¼ UT U for all TAAlg N:
5. Numerical radius isometries on atomic nest algebras Now we are in a position to give the main results of this paper promised in the introduction. Theorem 5.1. Let N and M be atomic nests on complex Hilbert spaces H and K; respectively. Assume that F : Alg N-Alg M is a weakly continuous and surjective
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map. Then F is a numerical radius isometry if and only if there exists a complex unit c and an operator RAAlg M such that one of the following statements is true: (1) There is a unitary operator (or, conjugate unitary operator) U : H-K satisfying UðNÞ ¼ M such that FðTÞ ¼ cUTU þ R for all TAAlg N: (2) There is a conjugate unitary operator (or, unitary operator) U : H-K satisfying UðN> Þ ¼ M such that FðTÞ ¼ cUT U þ R for all TAAlg N: (3) There exist H1 AN; K1 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K1 "K2 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M1 þ M2 ; respectively; there exist conjugate unitary operators (or, unitary operators) Us : Hs -Ks satisfying Us ðN> s Þ ¼ Ms ðs ¼ 1; 2Þ such that
FðTÞ ¼ c
U1 T1 U1
U1 T12 U2
0
U2 T2 U2
! þ R;
8T ¼
T1
T12
0
T2
! AAlg N:
(4) There exist H1 AN; K2 AM which determine the space decompositions H ¼ H1 "H2 ; K ¼ K2 "K1 ; and nest decompositions N ¼ N1 þ N2 ; M ¼ M2 þ M1 ; respectively; there exist unitary operators (or, conjugate unitary operators) Us : Hs -Ks satisfying Us ðNs Þ ¼ Ms ðs ¼ 1; 2Þ such that U2 T2 U2 FðTÞ ¼ c 0
U2 T12 U1 U1 T1 U1
! þ R;
8T ¼
T1 0
T12 T2
! AAlg N:
Theorem 5.2. Let N and M be two atomic nests on H and K; respectively, and F : DN -DM be a surjective map. Then F is a numerical radius isometry if and only if there exist space decompositions H ¼ X1 "X2 "X3 "X4 and K ¼ Y1 "Y2 "Y3 "Y4 with Xs and Ys spanned by some atoms of N and M; respectively; there exist unitary operators Us : Xs -Ys ðs ¼ 1; 2Þ; conjugate unitary operators Vt : Xt -Yt ðt ¼ 3; 4Þ; a unitary operator CAZðDM Þ (the center of DM ), and an operator RADM such that FðTÞ ¼ CðU1 T1 U1 "U2 T2 U2 "V3 T3 V3 "V4 T4 V4 Þ þ R for every TADN ; where Ts ¼ TjXs ðs ¼ 1; y; 4Þ: We remark that, by the observations after the proof of Theorem 3.11, every possible combinations of (1)–(4) in Theorem 5.1 may occur, according to the choice of N and M: To prove above results, we need a lemma which was obtained by Li and Sˇemrl in [12] for the case of upper triangular block matrix algebras. The idea of our proof is also borrowed from there. Lemma 5.3. Let N be an atomic nest on complex Hilbert space H: Suppose S; TAAlg N are real linearly independent such that wðsS þ tTÞp1
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whenever s; tAR satisfying s2 þ t2 p1: Then the following conditions are equivalent: (1) There exists a complex unit m such that ðS; TÞ ¼ mðI; 7iIÞ: (2) For any AAAlg N; there exist s; tAR with s2 þ t2 ¼ 1 such that wðsS þ tT þ AÞ ¼ 1 þ wðAÞ: Proof. ð1Þ ) ð2Þ is obvious. We need only prove ð2Þ ) ð1Þ: Assume that (2) holds. Let fHj gjAJ be the set of all atoms of N: For jAJ and any xAHj with jjxjj ¼ 1; set Ay ¼ ðcos y þ i sin yÞx#x; where yA½0; 2pÞ: Then there exist sy ; ty AR such that s2y þ t2y ¼ 1 and wðsy S þ ty T þ Ay Þ ¼ 1 þ wðAy Þ ¼ 2: Thus, there exists a unit vector sequence fxn gN n¼1 CH such that lim j/ðsy S þ ty T þ Ay Þxn ; xn Sj ¼ 2:
n-N
Hence, we must have lim j/ðsy S þ ty TÞxn ; xn Sj ¼ 1
n-N
and
lim j/ðAy xn ; xn Sj ¼ 1:
n-N
According to the space decomposition H ¼ ½x"½x> ; write xn ¼ an x"un : Then 1 ¼ limn-N j/ðAy xn ; xn Sj ¼ limn-N jan j2 j/Ay x; xSj ¼ limn-N jan j2 : So there is a complex number a with jaj ¼ 1 such that limn-N xn ¼ ax; and consequently, 2 ¼ j/ðsy S þ ty TÞx; xS þ /Ay x; xSj and j/ðsy S þ ty TÞx; xSj ¼ j/Ay x; xSj ¼ 1: Therefore, /ðsy S þ ty TÞx; xS ¼ /Ay x; xS ¼ cos y þ i sin y:
ð5:1Þ
Suppose /Sx; xS ¼ h1 þ ih2 ; /Tx; xS ¼ k1 þ ik2 ; where h1 ; h2 ; k1 ; k2 AR: Let ! h1 k1 C¼ AM2 ðRÞ: h2 k2 By equality (5.1), for any yA½0; 2pÞ; there exists a unit vector uy ¼ ðsy ; ty Þtr AR2 such that Cuy ¼ ðcos y; sin yÞtr : Hence C maps the unit ball in R2 onto itself. Thus C is an isometry on R2 and is of the form ! ! cos t sin t cos t sin t C¼ or : sin t cos t sin t cos t
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It follows that /Tx; xS ¼ 7i/Sx; xS and j/Sx; xSj ¼ 1 for each unit vector xAHj ; jAJ: Fix xAHj ; write /Sx; xS ¼ m and suppose that /Tx; xS ¼ im (the case /Tx; xS ¼ im may be treated similarly). Without loss of generality, we may assume m ¼ 1: With respect to the space decomposition H ¼ ½x"½x> ; S and T have the following matrix representations:
S¼
1
S12
S21
S22
! and
T ¼i
1
T12
T21
T22
! :
Note that for any self-adjoint operator AABðHÞ satisfying wðAÞ ¼ 1; if there exists Ax0 ¼ x0 and according some x0 AH with jjx0 jj ¼ 1 such that /Ax0 ; x0 S ¼ 1; then 1 0 > to the space decomposition H ¼ ½x0 "½x0 ; A ¼ : Thus we observe that, 0 A22 for every BABðHÞ with wðBÞ ¼ 1; if there is a unit vector x0 AH such that /Bx0 ; x0 S ¼ 1; then BþB has the direct sum decomposition in term of the space 2 1 B12 decomposition H ¼ ½x0 "½x0 > ; and consequently, B ¼ : B12 B22 For any yA½0; 2pÞ; let Zy ¼ ðcos y þ i sin yÞ1 ðcos yS þ sin yTÞ ¼ ðcos y þ i sin yÞ
1
cos y þ i sin y cos yS21 þ i sin yT21
cos yS12 þ i sin yT12 cos yS22 þ i sin yT22
! :
By (5.1), we can apply the above observation to Zy and get ðcos y þ i sin yÞ1 ðcos yS12 þ i sin yT12 Þ ¼ ððcos y þ i sin yÞ1 ðcos yS21 þ i sin yT21 ÞÞ ; ¼ S21 : so T12 ¼ S12 ¼ T21 For arbitrary unit vector yAH orthogonal to x satisfying y#yAAlg N; there is some kAJ so that yAHk : Without loss of generality, assume that jpk: For the sake of convenience, we write a ¼ /Sy; xS; l ¼ /Sy; yS; then /Sx; yS ¼ a% ; /Ty; yS ¼ dðiÞl; where dðiÞ ¼ 7i: For every yA½0; 2pÞ; we have
Zy;y ¼ P½x;y ðcos yS þ sin yTÞj½x;y ¼e
iy
1
a
a% eiy ðcos y þ dðiÞsin yÞl
! :
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Suppose aa0: If dðiÞ ¼ i; then there exists a yA½0; 2pÞ such that l% aei2y al% a% ei2y ; thus l% eiy Zy;y ¼
l% ei2y l% a% ei2y
l% aei2y
!
1
has numerical radius one and 12ðl% eiy Zy;y þ ðl% eiy Zy;y Þ Þ has an eigenvalue larger than one, which is a contradiction. Hence we have /Ty; yS ¼ il and Zy;y ¼ eiy
1
a
a%
l
! :
Let x be a complex unit such that ax ¼ jaj and let Bx ¼ xx#x þ xy#y þ 2x#y: Then W ðBx Þ is a circle disk centered at x with radius 1 and hence, wðBx Þ ¼ 2: By condition (2), there exists a yA½0; 2pÞ such that wðcos yS þ sin yT þ Bx Þ ¼ 3: A similar argument previous to (5.1) shows that there is a unit vector vAspanfx; yg such that 3 ¼ j/ðcos yS þ sin yTÞv; vS þ /Bx v; vSj ¼ j/ðcos yS þ sin yTÞv; vSj þ j/Bx v; vSj: Note that j/Bx v; vSj ¼ 2 if and only if v ¼ Zðx þ xyÞ for some ZAC with jZj ¼ Hence * ! + 1 þ l 1 a 1 ; 1 ¼ j/ðcos yS þ sinyTÞv; vSj ¼ v; v ¼ 2 2 a% l
pffiffi 2 2 :
and consequently, l ¼ 1: Therefore /Sy; yS ¼ /Sx; xS ¼ 1 and /Ty; yS ¼ i: Suppose a ¼ 0: Let xAf1; ig and Bx ¼ xðx#x þ y#yÞ þ 2x#y; then wðBx Þ ¼ 2: Now, similar to the previous argument, there exist s; tAR with s2 þ t2 ¼ 1 and a unit vector vAspanfx; yg such that 3 ¼ wðsS þ tT þ Bx Þ ¼ j/ðsS þ tT þ Bx Þv; vSj ¼ j/ðsS þ tTÞv; vS þ /Bx v; vSj ¼ j/ðsS þ tTÞv; vSj þ j/Bx v; vSj;
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hence /ðsS þ tTÞv; vS and /Bx v; vS have the same direction. If x ¼ 1; then /Bx v; vS ¼ 2 and 1 ¼ /ðsS þ tTÞv; vS ¼
ðs þ itÞ þ ðs7itÞl : 2
This entails that both complex units s þ it and ðs7itÞl are equal to 1, and consequently, s ¼ 1; t ¼ 0 and l ¼ 1: So /Sy; yS ¼ /Sx; xS ¼ 1: If x ¼ i; then /Bx v; vS ¼ 2i and i ¼ /ðsS þ tTÞv; vS ¼
ðs þ itÞ þ ðs7itÞ : 2
Thus s ¼ 0; t ¼ 1 and /Ty; yS ¼ i/Sy; yS ¼ i: Up to now we have proved, for arbitrary orthogonal unit vectors xAHj and yAHk with jpk; that /Sx; xS ¼ /Sy; yS with j/Sx; xSj ¼ 1 and /Ty; yS ¼ /Tx; xS ¼ i/Sx; xS: For every jAJ; let Pj denote the projection from H onto Hj : The convexity of numerical range, together with the fact that j/Sx; xSj ¼ 1 for all unit vectors xAHj ; implies that there exists a unit complex number mj such that Pj SPj ¼ mj Pj : Since, for any unit vectors xAHj and yAHk ; /Sx; xS ¼ /Sy; yS; there must be a constant m with jmj ¼ 1 such that mj ¼ m for all jAJ: We claim that S ¼ mI: Otherwise, there would exist some unit vectors xAHj and yAHk with jok so that m g g ¼ /Sy; xSa0: Thus S has a principal submatrix whose numerical radius 0 m is greater than jmj ¼ 1 ¼ wðSÞ; a contradiction. Thereby, for any unit vector xAHj ð8jAJÞ; we have /Tx; xS ¼ i/Sx; xS ¼ im and a similar argument implies T ¼ imI: & Proof of Theorem 5.1. We divide the proof into several steps. Step 1: For every TAAlg N; let CðTÞ ¼ FðTÞ Fð0Þ: Then C : Alg N-Alg M is a weakly continuous and surjective real linear map preserving numerical radius. It is obvious that Cð0Þ ¼ 0; C is weakly continuous, surjective and wðCðTÞÞ ¼ wðFðTÞ Fð0ÞÞ ¼ wðTÞ for every TAAlg N: Since the numerical radius is a norm, C is real linear from the Mazur–Ulam theorem [14] which states that every surjective isometry f satisfying fð0Þ ¼ 0 from a normed space onto another one is real linear. Step 2: There exists a complex unit c such that CðIÞ ¼ cI and CðiIÞ ¼ 7icI: For every AAAlg N; there exist some s; tAR with s2 þ t2 ¼ 1 such that wðsI þ itI þ AÞ ¼ 1 þ wðAÞ: Since C is real linear and preserves the numerical radius, we have wðsCðIÞ þ tCðiIÞ þ CðAÞÞ ¼ 1 þ wðCðAÞÞ for every AAAlg N: The surjectivity of C implies that ðCðIÞ; CðiIÞÞ satisfies condition (2) in Lemma 5.3, so there exists a cAC with jcj ¼ 1 such that CðIÞ ¼ cI and CðiIÞ ¼ 7icI: Without loss of generality, we assume that CðIÞ ¼ I and CðiIÞ ¼ 7iI: Hence, either CðlIÞ ¼ lI for all lAC; or CðlIÞ ¼ l% I for all lAC: Step 3: Assume that CðlIÞ ¼ lI for all lAC; then C preserves the closure of numerical range.
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For any TAAlg N; W ðTÞ is bounded and convex, so W ðTÞ is a compact convex subset in C: Assume that there exists a mAC such that mAW ðCðTÞÞ\W ðTÞ; then, there exists a circle with sufficiently large radius centered at a certain lAC such that W ðTÞ lies inside the circle, but m lies outside the circle. Hence, for any zAW ðTÞ; we have jz ljojm lj: Consequently, by the real linearity of C and the fact that CðlIÞ ¼ lI; one gets wðT lIÞojm ljpwðCðTÞ lIÞ ¼ wðCðT lIÞÞ ¼ wðT lIÞ; which is a contradiction. So W ðCðTÞÞDW ðTÞ and hence W ðCðTÞÞDW ðTÞ: Using the same argument to C1 ; we see that W ðTÞDW ðCðTÞÞ: It follows that, for all TAAlg N; W ðCðTÞÞ ¼ W ðTÞ; that is, C preserves the closure of numerical range. Step 4: Assume that for any lAC; CðlIÞ ¼ l% I: Let jðTÞ ¼ CðTÞ for every TAAlg N: Then j : Alg N-Alg M> is a weakly continuous and surjective additive map which preserves the closure of numerical range. It is clear that j is a weakly continuous and surjective additive map which preserves the numerical radius. Since jðlIÞ ¼ lI for each lAC; by the proof of Step 3, we have j : Alg N-Alg M> preserves the closure of numerical range. Now, applying Theorem 4.1 to C in Step 3 and to j in Step 4, it is easily seen that F has one of the eight forms stated in Theorem 5.1 and the proof is finished. To make the point clearer, we assume, for example, that j has form (1) in Theorem 4.1. That is, there exists a unitary operator UABðH; KÞ satisfying UðNÞ ¼ M> such that jðTÞ ¼ UTU for every TAAlg N: Then UðN> Þ ¼ M and CðTÞ ¼ UT U for every TAAlg N; which is one of two forms stated in (2) of Theorem 5.1. & We remark that, by the observations after the proof of Theorem 3.11, all possible combinations of forms (1)–(4) in Theorem 5.1 may occur. Particularly, in many cases, the nest N may not be unitarily equivalent to the nest M> ; or N and M may not have the decompositions described in (3) or (4) of Theorem 5.1. Thus the classification of numerical radius isometries from Alg N onto Alg M may sometimes have a neater expression. The following corollary is an immediate consequence of Theorems 3.11 and 5.1, which omits the assumption of weak continuity. Corollary 5.4. Let N be an atomic nest on Hilbert space H with the order type o þ 1 or 1 þ o and let F : Alg N-Alg N be a surjective map. Assume that every atom of N is finite dimensional. Then F is a numerical radius isometry if and only if there exists a unitary operator or conjugate unitary operator U on H satisfying UðNÞ ¼ N; an operator RAAlg N and a complex unit c such that FðTÞ ¼ cUTU þ R holds for every TAAlg N: In other words, Corollary 5.4 says that, if N be an atomic nest on Hilbert space H with the order type o þ 1 or 1 þ o and if its every atom is of finite dimension, then
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the set of all numerical radius isometries on Alg N is a group generated by the following simple maps: (1) T/UTU ; where U : H-H is a unitary or conjugate unitary operator satisfying UðNÞ ¼ N; (2) T/cT; where cAC and jcj ¼ 1; (3) T/T þ R; where RAAlg N: Next, we turn to the proof of Theorem 5.2. Proof of Theorem 5.2. Let fHj j jAJg and fKk j kAKg be the sets of all atoms of N and M; respectively. Then DN ¼ "jAJ BðHj Þ and DM ¼ "kAK BðKk Þ: Notice that CAZðDM Þ if and only if there exists a set fmk gkAK of complex numbers such that C ¼ "kAK mk Ik : Thus, for every GADM ; we have wðCGÞ ¼ wðGÞ if C is also unitary. So, the sufficiency of the condition is obvious and we need only verify the necessity. Assume that F is a numerical radius isometry. For each TADN ; let CðTÞ ¼ FðTÞ Fð0Þ; then C is a real linear map preserving the numerical radius. Claim. Suppose S; TADM are real linearly independent such that wðsS þ tTÞp1 whenever s; tAR satisfying s2 þ t2 p1: Then the following conditions are equivalent: (i) There exist complex units fmk j kAKg such that ðS; TÞ ¼ " mk ðIk ; 7iIk Þ; where Ik denotes the identity on Kk : kAK
(ii) For any AADM ; there exist s; tAR with s2 þ t2 ¼ 1 such that wðsS þ tT þ AÞ ¼ 1 þ wðAÞ: To prove the claim, write S ¼ "kAK Sk and T ¼ "kAK Tk ; where Tk ; Sk ABðKk Þ ðkAKÞ: Suppose that (i) holds. Let A ¼ "kAK Ak ADM be arbitrary. It is clear that, for any e40; there exists a kAK such that wðAÞowðAk Þ þ e: Using Lemma 5.3 for Ak ; we get that there exist s; tAR with s2 þ t2 ¼ 1 such that 1 þ wðAÞo 1 þ wðAk Þ þ e ¼ wðsSk þ tTk þ Ak Þ þ e p wðsS þ tT þ AÞ þ ep1 þ wðAÞ þ e: Let e-0; we see that condition (ii) holds. Conversely, suppose that (ii) holds. For any k0 AK; take Aðk0 Þ ¼ "kAK Ak ADM ; where Ak ¼ 0 if kak0 : Then 1 þ wðAk0 Þ ¼ 1 þ wðAðk0 Þ Þ ¼ wðsS þ tT þ Aðk0 Þ Þ ¼ wðsSk0 þ tTk0 þ Aðk0 Þ Þp1 þ wðAk0 Þ; so wðsSk0 þ tTk0 þ Aðk0 Þ Þ ¼ 1 þ wðAk0 Þ: Now by Lemma 5.3, there exists a complex unit mk0 such that ðSk0 ; Tk0 Þ ¼ mk0 ðIk0 ; 7iIk0 Þ: Therefore we have ðiiÞ ) ðiÞ: Hence Claim holds.
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Since for any BADN ; there exist s; tAR with s2 þ t2 ¼ 1 such that wðsI þ itI þ BÞ ¼ 1 þ wðBÞ; we have wðsCðIÞ þ tCðiIÞ þ CðBÞÞ ¼ 1 þ wðCðBÞÞ by the numerical radius preservation and real linearity of C: Now Claim, together with the surjectivity of C; implies that there exist complex units mk ð8kAKÞ such that ðCðIÞ; CðiIÞÞ ¼ "kAK mk ðIk ; 7iIk Þ: Let jðTÞ ¼ CðIÞ1 CðTÞ for every TADN : Since CðTÞ has the form CðTÞ ¼ "kAk Ek ; we have jðTÞ ¼ "kAk mk Ek and wðjðTÞÞ ¼ supkAk fwðmk Ek Þg ¼ wðCðTÞÞ ¼ wðTÞ: Hence j : DN -DM is a unital real linear map preserving the numerical radius. Let H 1 ¼ fkAK j jðiIÞk ¼ iIk g and H 2 ¼ fkAK j jðiIÞk ¼ iIk g: Then H 1 ,H 2 ¼ K and jðiIÞ ¼ ð"kAH 1 iIk Þ"ð"kAH 2 ðiIk ÞÞ: Define p : DM -DM by pð"kAK Gk Þ ¼ ð"kAH 1 Gk Þ"ð"kAH 2 Gk Þ for every "kAK Gk ADM : It is obvious that p is a real linear bijection preserving the numerical radius. Let c ¼ p 3 j; then c : DN -DM is a real linear surjective map which preserves the numerical radius and cðlIÞ ¼ lI for each lAC: A similar argument just as in Step 3 of the proof of Theorem 5.1 implies that c preserves the closure of numerical range. Now it follows from Theorem 2.3 and its proof that there exists a 1-1 and onto map y : J-K; there exist subsets J01 and J02 of J with J ¼ J01 ,J02 ; and there exist unitary operators Uj ABðHj ; Kyð jÞ Þ if jAJ01 ; conjugate unitary operators Vj : Hj -Kyð jÞ if jAJ02 ; such that for every T ¼ "jAJ Tj ADN ; ! ! cðTÞ ¼
" Uy1 ðkÞ Ty1 ðkÞ Uy1 ðkÞ "
kAK01
" Vy1 ðkÞ Ty1 ðkÞ Vy1 ðkÞ ;
kAK02
where K0s ¼ yðJ0s Þ; s ¼ 1; 2: Let Ks ¼ K01 -H s and K2þt ¼ K02 -H t ; s; t ¼ 1; 2: Then for every T ¼ "jAJ Tj ADN ; we have jðTÞ ¼ " Uy1 ðkÞ Ty1 ðkÞ Uy1 ðkÞ " " Uy1 ðkÞ Ty1 ðkÞ Uy1 ðkÞ kAK1 kAK2 " " Vy1 ðkÞ Ty1 ðkÞ Vy1 ðkÞ " " Vy1 ðkÞ Ty1 ðkÞ Vy1 ðkÞ : kAK3
kAK4
Put C ¼ CðIÞ and R ¼ Fð0Þ; then we see that F has the desired form.
&
References [1] Z.F. Bai, J.C. Hou, Numerical radius distance preserving maps on BðHÞ; Proc. Amer. Math. Soc., to appear. [2] J.L. Cui, J.C. Hou, Linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest, Integer Equation and Operator Theory, to appear. [3] K.R. Davidson, Nest Algebra, in: Ritman Research Notes in Mathematics, Vol. 191, Longman, London, New York, 1988. [4] G. Frobenius, Uber die darstellung der endlichen gruppen durch lineare subsitutionen, I., Sitzungberichte Koniglich Preussischen Akad. Wissenschaften Berlin, 1897, pp. 994–1015.
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[5] P.R. Halmos, A Hilbert Space Problem Book, 2nd Edition, Springer, New York, 1982. [6] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991. [7] L.K. Hua, Geometry of matrices I. Generalizations of von staudt’s theorem, Trans. Amer. Math. Soc. 57 (1945) 441–481. [8] L.K. Hua, Geometry of matrices II. Study of involutions in the geometry of symmetric matrices, Trans. Amer. Math. Soc. 61 (1947) 193–228. [9] L.K. Hua, Geometry of matrices III. Fundamental theorems in the geometry of symmetric matrices, Trans. Amer. Math. Soc. 61 (1947) 229–255. [10] L.K. Hua, Geometry of symmetric matrices over any field with characteristic other than two, Ann. Math. 50 (1949) 8–31. [11] G. Lesnjak, Additive preserves of numerical range, Linear Algebra Appl. 345 (2002) 235–253. [12] C.K. Li, P. Semrl, Numerical radius isometries, Linear and Multilinear Algebras 50 (2002) 307–314. [13] C.K. Li, N.K. Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162–164 (1992) 217–235. [14] S. Mazur, S. Ulam, Sur les transformations isom’etriques d’espaces vectoriels norm’es, C. R. Acad. Sci. Paris 194 (1932) 946–948. [15] Z.X. Wan, Geometry of Matrices, World Scientific, Singapore, 1996.